An equation is a mathematical statement that shows the equality of two expressions . To solve an equation , you need to find the value of the variable that makes the equation true.

## Linear Equations

A linear equation is an equation of the form ax + b = c, where a, b, and c are constants , and x is the variable. To solve a linear equation , isolate the variable on one side of the equation by performing inverse operations .

## Quadratic Equations

A quadratic equation is an equation of the form ax^2 + bx + c = 0, where a, b, and c are constants , and x is the variable. To solve a quadratic equation , you can use methods like factoring , completing the square , or using the quadratic formula .

## Inequalities

An inequality is a mathematical statement that shows the relationship between two expressions , where one is greater than , less than , greater than or equal to , or less than or equal to the other.

## Linear Inequalities

A linear inequality is an inequality of the form ax + b c, where a, b, and c are constants , and x is the variable. To solve a linear inequality , you can use similar methods as solving linear equations , but you also need to consider the direction of the inequality sign.

## Quadratic Inequalities

A quadratic inequality is an inequality of the form ax^2 + bx + c 0. To solve a quadratic inequality , you can use techniques such as graphing or testing intervals to determine the solution set.

- Practice solving equations and inequalities of different types to familiarize yourself with various problem-solving methods.
- Understand the properties of equality and inequality , such as the addition and multiplication properties, which allow you to perform operations on both sides of an equation or inequality .
- Pay attention to special cases when solving equations and inequalities, such as division by zero or taking the square root of both sides .
- Use visual aids and graphs to understand the solutions to inequalities, especially quadratic inequalities.
- Review the steps for solving different types of equations and inequalities to build a strong problem-solving approach.

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## Unit 7: Equations & inequalities

About this unit.

Solving equations is a superpower. It means we can model a situation with an equation in any way that makes sense to us, even with an unknown value in the middle. Inequalities are for situations with many true options, like how many pages I can send in my letter using just 1 stamp.

## Algebraic equations basics

- Variables, expressions, & equations (Opens a modal)
- Testing solutions to equations (Opens a modal)
- Intro to equations (Opens a modal)
- Testing solutions to equations Get 5 of 7 questions to level up!

## One-step equations intuition

- Same thing to both sides of equations (Opens a modal)
- Representing a relationship with an equation (Opens a modal)
- Dividing both sides of an equation (Opens a modal)
- One-step equations intuition (Opens a modal)
- Identify equations from visual models (tape diagrams) Get 3 of 4 questions to level up!
- Identify equations from visual models (hanger diagrams) Get 3 of 4 questions to level up!
- Solve equations from visual models Get 3 of 4 questions to level up!

## One-step addition & subtraction equations

- One-step addition & subtraction equations (Opens a modal)
- One-step addition equation (Opens a modal)
- One-step addition & subtraction equations: fractions & decimals (Opens a modal)
- One-step addition & subtraction equations Get 5 of 7 questions to level up!
- One-step addition & subtraction equations: fractions & decimals Get 5 of 7 questions to level up!

## One-step multiplication and division equations

- One-step division equations (Opens a modal)
- One-step multiplication equations (Opens a modal)
- One-step multiplication & division equations (Opens a modal)
- One-step multiplication & division equations: fractions & decimals (Opens a modal)
- One-step multiplication equations: fractional coefficients (Opens a modal)
- One-step multiplication & division equations Get 5 of 7 questions to level up!
- One-step multiplication & division equations: fractions & decimals Get 5 of 7 questions to level up!

## Finding mistakes in one-step equations

- Finding mistakes in one-step equations (Opens a modal)
- Find the mistake in one-step equations Get 3 of 4 questions to level up!

## One-step equation word problems

- Modeling with one-step equations (Opens a modal)
- Translate one-step equations and solve Get 3 of 4 questions to level up!
- Model with one-step equations Get 3 of 4 questions to level up!
- Model with one-step equations and solve Get 3 of 4 questions to level up!

## Intro to inequalities with variables

- Testing solutions to inequalities (Opens a modal)
- Plotting inequalities (Opens a modal)
- Plotting an inequality example (Opens a modal)
- Inequalities word problems (Opens a modal)
- Graphing inequalities review (Opens a modal)
- Testing solutions to inequalities (basic) Get 3 of 4 questions to level up!
- Graphing basic inequalities Get 3 of 4 questions to level up!
- Inequality from graph Get 3 of 4 questions to level up!
- Plotting inequalities Get 3 of 4 questions to level up!
- Inequalities word problems Get 3 of 4 questions to level up!

## Dependent and independent variables

- Dependent & independent variables (Opens a modal)
- Dependent & independent variables: graphing (Opens a modal)
- Dependent and independent variables review (Opens a modal)
- Independent versus dependent variables Get 3 of 4 questions to level up!
- Tables from equations with 2 variables Get 3 of 4 questions to level up!
- Match equations to coordinates on a graph Get 3 of 4 questions to level up!

## Analyzing relationships between variables

- Writing equations for relationships between quantities (Opens a modal)
- Analyzing relationships between variables (Opens a modal)
- Equations and inequalities FAQ (Opens a modal)
- Relationships between quantities in equations Get 3 of 4 questions to level up!
- Analyze relationships between variables Get 3 of 4 questions to level up!

## 5.2 Solving Systems of Equations by Substitution

Learning objectives.

By the end of this section, you will be able to:

- Solve a system of equations by substitution
- Solve applications of systems of equations by substitution

## Be Prepared 5.4

Before you get started, take this readiness quiz.

Simplify −5 ( 3 − x ) −5 ( 3 − x ) . If you missed this problem, review Example 1.136 .

## Be Prepared 5.5

Simplify 4 − 2 ( n + 5 ) 4 − 2 ( n + 5 ) . If you missed this problem, review Example 1.123 .

## Be Prepared 5.6

Solve for y y : 8 y − 8 = 32 − 2 y 8 y − 8 = 32 − 2 y If you missed this problem, review Example 2.34 .

## Be Prepared 5.7

Solve for x x : 3 x − 9 y = −3 3 x − 9 y = −3 If you missed this problem, review Example 2.65 .

Solving systems of linear equations by graphing is a good way to visualize the types of solutions that may result. However, there are many cases where solving a system by graphing is inconvenient or imprecise. If the graphs extend beyond the small grid with x and y both between −10 and 10, graphing the lines may be cumbersome. And if the solutions to the system are not integers, it can be hard to read their values precisely from a graph.

In this section, we will solve systems of linear equations by the substitution method.

Solve a System of Equations by Substitution

We will use the same system we used first for graphing.

We will first solve one of the equations for either x or y . We can choose either equation and solve for either variable—but we’ll try to make a choice that will keep the work easy.

Then we substitute that expression into the other equation. The result is an equation with just one variable—and we know how to solve those!

After we find the value of one variable, we will substitute that value into one of the original equations and solve for the other variable. Finally, we check our solution and make sure it makes both equations true.

We’ll fill in all these steps now in Example 5.13 .

## Example 5.13

How to solve a system of equations by substitution.

Solve the system by substitution. { 2 x + y = 7 x − 2 y = 6 { 2 x + y = 7 x − 2 y = 6

## Try It 5.25

Solve the system by substitution. { −2 x + y = −11 x + 3 y = 9 { −2 x + y = −11 x + 3 y = 9

## Try It 5.26

Solve the system by substitution. { x + 3 y = 10 4 x + y = 18 { x + 3 y = 10 4 x + y = 18

## Solve a system of equations by substitution.

- Step 1. Solve one of the equations for either variable.
- Step 2. Substitute the expression from Step 1 into the other equation.
- Step 3. Solve the resulting equation.
- Step 4. Substitute the solution in Step 3 into one of the original equations to find the other variable.
- Step 5. Write the solution as an ordered pair.
- Step 6. Check that the ordered pair is a solution to both original equations.

If one of the equations in the system is given in slope–intercept form, Step 1 is already done! We’ll see this in Example 5.14 .

## Example 5.14

Solve the system by substitution.

{ x + y = −1 y = x + 5 { x + y = −1 y = x + 5

The second equation is already solved for y . We will substitute the expression in place of y in the first equation.

The second equation is already solved for . We will substitute into the first equation. | |

Replace the with + 5. | |

Solve the resulting equation for . | |

Substitute = −3 into = + 5 to find . | |

The ordered pair is (−3, 2). | |

Check the ordered pair in both equations: | |

The solution is (−3, 2). |

## Try It 5.27

Solve the system by substitution. { x + y = 6 y = 3 x − 2 { x + y = 6 y = 3 x − 2

## Try It 5.28

Solve the system by substitution. { 2 x − y = 1 y = −3 x − 6 { 2 x − y = 1 y = −3 x − 6

If the equations are given in standard form, we’ll need to start by solving for one of the variables. In this next example, we’ll solve the first equation for y .

## Example 5.15

Solve the system by substitution. { 3 x + y = 5 2 x + 4 y = −10 { 3 x + y = 5 2 x + 4 y = −10

We need to solve one equation for one variable. Then we will substitute that expression into the other equation.

Solve for . Substitute into the other equation. | |

Replace the with −3 + 5. | |

Solve the resulting equation for . | |

| |

Substitute = 3 into 3 + = 5 to find . | |

| |

The ordered pair is (3, −4). | |

Check the ordered pair in both equations: | |

The solution is (3, −4). |

## Try It 5.29

Solve the system by substitution. { 4 x + y = 2 3 x + 2 y = −1 { 4 x + y = 2 3 x + 2 y = −1

## Try It 5.30

Solve the system by substitution. { − x + y = 4 4 x − y = 2 { − x + y = 4 4 x − y = 2

In Example 5.15 it was easiest to solve for y in the first equation because it had a coefficient of 1. In Example 5.16 it will be easier to solve for x .

## Example 5.16

Solve the system by substitution. { x − 2 y = −2 3 x + 2 y = 34 { x − 2 y = −2 3 x + 2 y = 34

We will solve the first equation for x x and then substitute the expression into the second equation.

Solve for . Substitute into the other equation. | |

Replace the with 2 − 2. | |

Solve the resulting equation for . | |

Substitute = 5 into − 2 = −2 to find . | |

The ordered pair is (8, 5). | |

Check the ordered pair in both equations: | |

The solution is (8, 5). |

## Try It 5.31

Solve the system by substitution. { x − 5 y = 13 4 x − 3 y = 1 { x − 5 y = 13 4 x − 3 y = 1

## Try It 5.32

Solve the system by substitution. { x − 6 y = −6 2 x − 4 y = 4 { x − 6 y = −6 2 x − 4 y = 4

When both equations are already solved for the same variable, it is easy to substitute!

## Example 5.17

Solve the system by substitution. { y = −2 x + 5 y = 1 2 x { y = −2 x + 5 y = 1 2 x

Since both equations are solved for y , we can substitute one into the other.

Substitute for in the first equation. | |

Replace the with | |

Solve the resulting equation. Start by clearing the fraction. | |

Solve for . | |

Substitute = 2 into = to find . | |

The ordered pair is (2,1). | |

Check the ordered pair in both equations: | |

The solution is (2,1). |

## Try It 5.33

Solve the system by substitution. { y = 3 x − 16 y = 1 3 x { y = 3 x − 16 y = 1 3 x

## Try It 5.34

Solve the system by substitution. { y = − x + 10 y = 1 4 x { y = − x + 10 y = 1 4 x

Be very careful with the signs in the next example.

## Example 5.18

Solve the system by substitution. { 4 x + 2 y = 4 6 x − y = 8 { 4 x + 2 y = 4 6 x − y = 8

We need to solve one equation for one variable. We will solve the first equation for y .

Solve the first equation for . | |

Substitute −2 + 2 for in the second equation. | |

Replace the with −2 + 2. | |

Solve the equation for . | |

| |

Substitute into 4 + 2 = 4 to find . | |

The ordered pair is | |

Check the ordered pair in both equations. | |

The solution is |

## Try It 5.35

Solve the system by substitution. { x − 4 y = −4 −3 x + 4 y = 0 { x − 4 y = −4 −3 x + 4 y = 0

## Try It 5.36

Solve the system by substitution. { 4 x − y = 0 2 x − 3 y = 5 { 4 x − y = 0 2 x − 3 y = 5

In Example 5.19 , it will take a little more work to solve one equation for x or y .

## Example 5.19

Solve the system by substitution. { 4 x − 3 y = 6 15 y − 20 x = −30 { 4 x − 3 y = 6 15 y − 20 x = −30

We need to solve one equation for one variable. We will solve the first equation for x .

Solve the first equation for . | |

Substitute for in the second equation. | |

Replace the with | |

Solve for . | |

Since 0 = 0 is a true statement, the system is consistent. The equations are dependent. The graphs of these two equations would give the same line. The system has infinitely many solutions.

## Try It 5.37

Solve the system by substitution. { 2 x − 3 y = 12 −12 y + 8 x = 48 { 2 x − 3 y = 12 −12 y + 8 x = 48

## Try It 5.38

Solve the system by substitution. { 5 x + 2 y = 12 −4 y − 10 x = −24 { 5 x + 2 y = 12 −4 y − 10 x = −24

Look back at the equations in Example 5.19 . Is there any way to recognize that they are the same line?

Let’s see what happens in the next example.

## Example 5.20

Solve the system by substitution. { 5 x − 2 y = −10 y = 5 2 x { 5 x − 2 y = −10 y = 5 2 x

The second equation is already solved for y , so we can substitute for y in the first equation.

Substitute for in the first equation. | |

Replace the with | |

Solve for . | |

Since 0 = −10 is a false statement the equations are inconsistent. The graphs of the two equation would be parallel lines. The system has no solutions.

## Try It 5.39

Solve the system by substitution. { 3 x + 2 y = 9 y = − 3 2 x + 1 { 3 x + 2 y = 9 y = − 3 2 x + 1

## Try It 5.40

Solve the system by substitution. { 5 x − 3 y = 2 y = 5 3 x − 4 { 5 x − 3 y = 2 y = 5 3 x − 4

Solve Applications of Systems of Equations by Substitution

We’ll copy here the problem solving strategy we used in the Solving Systems of Equations by Graphing section for solving systems of equations. Now that we know how to solve systems by substitution, that’s what we’ll do in Step 5.

## How to use a problem solving strategy for systems of linear equations.

- Step 1. Read the problem. Make sure all the words and ideas are understood.
- Step 2. Identify what we are looking for.
- Step 3. Name what we are looking for. Choose variables to represent those quantities.
- Step 4. Translate into a system of equations.
- Step 5. Solve the system of equations using good algebra techniques.
- Step 6. Check the answer in the problem and make sure it makes sense.
- Step 7. Answer the question with a complete sentence.

Some people find setting up word problems with two variables easier than setting them up with just one variable. Choosing the variable names is easier when all you need to do is write down two letters. Think about this in the next example—how would you have done it with just one variable?

## Example 5.21

The sum of two numbers is zero. One number is nine less than the other. Find the numbers.

the problem. | |

what we are looking for. | We are looking for two numbers. |

what we are looking for. | Let the first number Let the second number |

into a system of equations. | The sum of two numbers is zero. |

One number is nine less than the other. | |

The system is: | |

the system of equations. We will use substitution since the second equation is solved for . | |

Substitute − 9 for in the first equation. | |

Solve for . | |

Substitute into the second equation and then solve for . | |

the answer in the problem. | Do these numbers make sense in the problem? We will leave this to you! |

the question. | The numbers are and |

## Try It 5.41

The sum of two numbers is 10. One number is 4 less than the other. Find the numbers.

## Try It 5.42

The sum of two number is −6. One number is 10 less than the other. Find the numbers.

In the Example 5.22 , we’ll use the formula for the perimeter of a rectangle, P = 2 L + 2 W .

## Example 5.22

The perimeter of a rectangle is 88. The length is five more than twice the width. Find the length and the width.

the problem. | |

what you are looking for. | We are looking for the length and width. |

what we are looking for. | Let the length the width |

into a system of equations. | The perimeter of a rectangle is 88. |

2 + 2 = | |

The length is five more than twice the width. | |

The system is: | |

the system of equations. We will use substitution since the second equation is solved for . Substitute 2 + 5 for in the first equation. | |

Solve for . | |

Substitute = 13 into the second equation and then solve for . | |

the answer in the problem. | Does a rectangle with length 31 and width 13 have perimeter 88? Yes. |

the equation. | The length is 31 and the width is 13. |

## Try It 5.43

The perimeter of a rectangle is 40. The length is 4 more than the width. Find the length and width of the rectangle.

## Try It 5.44

The perimeter of a rectangle is 58. The length is 5 more than three times the width. Find the length and width of the rectangle.

For Example 5.23 we need to remember that the sum of the measures of the angles of a triangle is 180 degrees and that a right triangle has one 90 degree angle.

## Example 5.23

The measure of one of the small angles of a right triangle is ten more than three times the measure of the other small angle. Find the measures of both angles.

We will draw and label a figure.

the problem. | |

what you are looking for. | We are looking for the measures of the angles. |

what we are looking for. | Let the measure of the 1 angle the measure of the 2 angle |

into a system of equations. | The measure of one of the small angles of a right triangle is ten more than three times the measure of the other small angle. |

The sum of the measures of the angles of a triangle is 180. | |

The system is: | |

the system of equations. We will use substitution since the first equation is solved for . | |

Substitute 3 + 10 for in the second equation. | |

Solve for . | |

Substitute = 20 into the first equation and then solve for . | |

the answer in the problem. | We will leave this to you! |

the question. | The measures of the small angles are 20 and 70. |

## Try It 5.45

The measure of one of the small angles of a right triangle is 2 more than 3 times the measure of the other small angle. Find the measure of both angles.

## Try It 5.46

The measure of one of the small angles of a right triangle is 18 less than twice the measure of the other small angle. Find the measure of both angles.

## Example 5.24

Heather has been offered two options for her salary as a trainer at the gym. Option A would pay her $25,000 plus $15 for each training session. Option B would pay her $10,000 + $40 for each training session. How many training sessions would make the salary options equal?

the problem. | |

what you are looking for. | We are looking for the number of training sessions that would make the pay equal. |

what we are looking for. | Let Heather’s salary. the number of training sessions |

into a system of equations. | Option A would pay her $25,000 plus $15 for each training session. |

Option B would pay her $10,000 + $40 for each training session | |

The system is: | |

the system of equations. We will use substitution. | |

Substitute 25,000 + 15 for in the second equation. | |

Solve for . | |

the answer. | Are 600 training sessions a year reasonable? Are the two options equal when = 600? |

the question. | The salary options would be equal for 600 training sessions. |

## Try It 5.47

Geraldine has been offered positions by two insurance companies. The first company pays a salary of $12,000 plus a commission of $100 for each policy sold. The second pays a salary of $20,000 plus a commission of $50 for each policy sold. How many policies would need to be sold to make the total pay the same?

## Try It 5.48

Kenneth currently sells suits for company A at a salary of $22,000 plus a $10 commission for each suit sold. Company B offers him a position with a salary of $28,000 plus a $4 commission for each suit sold. How many suits would Kenneth need to sell for the options to be equal?

Access these online resources for additional instruction and practice with solving systems of equations by substitution.

- Instructional Video-Solve Linear Systems by Substitution
- Instructional Video-Solve by Substitution

## Section 5.2 Exercises

Practice makes perfect.

In the following exercises, solve the systems of equations by substitution.

{ 2 x + y = −4 3 x − 2 y = −6 { 2 x + y = −4 3 x − 2 y = −6

{ 2 x + y = −2 3 x − y = 7 { 2 x + y = −2 3 x − y = 7

{ x − 2 y = −5 2 x − 3 y = −4 { x − 2 y = −5 2 x − 3 y = −4

{ x − 3 y = −9 2 x + 5 y = 4 { x − 3 y = −9 2 x + 5 y = 4

{ 5 x − 2 y = −6 y = 3 x + 3 { 5 x − 2 y = −6 y = 3 x + 3

{ −2 x + 2 y = 6 y = −3 x + 1 { −2 x + 2 y = 6 y = −3 x + 1

{ 2 x + 3 y = 3 y = − x + 3 { 2 x + 3 y = 3 y = − x + 3

{ 2 x + 5 y = −14 y = −2 x + 2 { 2 x + 5 y = −14 y = −2 x + 2

{ 2 x + 5 y = 1 y = 1 3 x − 2 { 2 x + 5 y = 1 y = 1 3 x − 2

{ 3 x + 4 y = 1 y = − 2 5 x + 2 { 3 x + 4 y = 1 y = − 2 5 x + 2

{ 3 x − 2 y = 6 y = 2 3 x + 2 { 3 x − 2 y = 6 y = 2 3 x + 2

{ −3 x − 5 y = 3 y = 1 2 x − 5 { −3 x − 5 y = 3 y = 1 2 x − 5

{ 2 x + y = 10 − x + y = −5 { 2 x + y = 10 − x + y = −5

{ −2 x + y = 10 − x + 2 y = 16 { −2 x + y = 10 − x + 2 y = 16

{ 3 x + y = 1 −4 x + y = 15 { 3 x + y = 1 −4 x + y = 15

{ x + y = 0 2 x + 3 y = −4 { x + y = 0 2 x + 3 y = −4

{ x + 3 y = 1 3 x + 5 y = −5 { x + 3 y = 1 3 x + 5 y = −5

{ x + 2 y = −1 2 x + 3 y = 1 { x + 2 y = −1 2 x + 3 y = 1

{ 2 x + y = 5 x − 2 y = −15 { 2 x + y = 5 x − 2 y = −15

{ 4 x + y = 10 x − 2 y = −20 { 4 x + y = 10 x − 2 y = −20

{ y = −2 x − 1 y = − 1 3 x + 4 { y = −2 x − 1 y = − 1 3 x + 4

{ y = x − 6 y = − 3 2 x + 4 { y = x − 6 y = − 3 2 x + 4

{ y = 2 x − 8 y = 3 5 x + 6 { y = 2 x − 8 y = 3 5 x + 6

{ y = − x − 1 y = x + 7 { y = − x − 1 y = x + 7

{ 4 x + 2 y = 8 8 x − y = 1 { 4 x + 2 y = 8 8 x − y = 1

{ − x − 12 y = −1 2 x − 8 y = −6 { − x − 12 y = −1 2 x − 8 y = −6

{ 15 x + 2 y = 6 −5 x + 2 y = −4 { 15 x + 2 y = 6 −5 x + 2 y = −4

{ 2 x − 15 y = 7 12 x + 2 y = −4 { 2 x − 15 y = 7 12 x + 2 y = −4

{ y = 3 x 6 x − 2 y = 0 { y = 3 x 6 x − 2 y = 0

{ x = 2 y 4 x − 8 y = 0 { x = 2 y 4 x − 8 y = 0

{ 2 x + 16 y = 8 − x − 8 y = −4 { 2 x + 16 y = 8 − x − 8 y = −4

{ 15 x + 4 y = 6 −30 x − 8 y = −12 { 15 x + 4 y = 6 −30 x − 8 y = −12

{ y = −4 x 4 x + y = 1 { y = −4 x 4 x + y = 1

{ y = − 1 4 x x + 4 y = 8 { y = − 1 4 x x + 4 y = 8

{ y = 7 8 x + 4 −7 x + 8 y = 6 { y = 7 8 x + 4 −7 x + 8 y = 6

{ y = − 2 3 x + 5 2 x + 3 y = 11 { y = − 2 3 x + 5 2 x + 3 y = 11

In the following exercises, translate to a system of equations and solve.

The sum of two numbers is 15. One number is 3 less than the other. Find the numbers.

The sum of two numbers is 30. One number is 4 less than the other. Find the numbers.

The sum of two numbers is −26. One number is 12 less than the other. Find the numbers.

The perimeter of a rectangle is 50. The length is 5 more than the width. Find the length and width.

The perimeter of a rectangle is 60. The length is 10 more than the width. Find the length and width.

The perimeter of a rectangle is 58. The length is 5 more than three times the width. Find the length and width.

The perimeter of a rectangle is 84. The length is 10 more than three times the width. Find the length and width.

The measure of one of the small angles of a right triangle is 14 more than 3 times the measure of the other small angle. Find the measure of both angles.

The measure of one of the small angles of a right triangle is 26 more than 3 times the measure of the other small angle. Find the measure of both angles.

The measure of one of the small angles of a right triangle is 15 less than twice the measure of the other small angle. Find the measure of both angles.

The measure of one of the small angles of a right triangle is 45 less than twice the measure of the other small angle. Find the measure of both angles.

Maxim has been offered positions by two car dealers. The first company pays a salary of $10,000 plus a commission of $1,000 for each car sold. The second pays a salary of $20,000 plus a commission of $500 for each car sold. How many cars would need to be sold to make the total pay the same?

Jackie has been offered positions by two cable companies. The first company pays a salary of $ 14,000 plus a commission of $100 for each cable package sold. The second pays a salary of $20,000 plus a commission of $25 for each cable package sold. How many cable packages would need to be sold to make the total pay the same?

Amara currently sells televisions for company A at a salary of $17,000 plus a $100 commission for each television she sells. Company B offers her a position with a salary of $29,000 plus a $20 commission for each television she sells. How many televisions would Amara need to sell for the options to be equal?

Mitchell currently sells stoves for company A at a salary of $12,000 plus a $150 commission for each stove he sells. Company B offers him a position with a salary of $24,000 plus a $50 commission for each stove he sells. How many stoves would Mitchell need to sell for the options to be equal?

## Everyday Math

When Gloria spent 15 minutes on the elliptical trainer and then did circuit training for 30 minutes, her fitness app says she burned 435 calories. When she spent 30 minutes on the elliptical trainer and 40 minutes circuit training she burned 690 calories. Solve the system { 15 e + 30 c = 435 30 e + 40 c = 690 { 15 e + 30 c = 435 30 e + 40 c = 690 for e e , the number of calories she burns for each minute on the elliptical trainer, and c c , the number of calories she burns for each minute of circuit training.

Stephanie left Riverside, California, driving her motorhome north on Interstate 15 towards Salt Lake City at a speed of 56 miles per hour. Half an hour later, Tina left Riverside in her car on the same route as Stephanie, driving 70 miles per hour. Solve the system { 56 s = 70 t s = t + 1 2 { 56 s = 70 t s = t + 1 2 .

- ⓐ for t t to find out how long it will take Tina to catch up to Stephanie.
- ⓑ what is the value of s s , the number of hours Stephanie will have driven before Tina catches up to her?

## Writing Exercises

Solve the system of equations { x + y = 10 x − y = 6 { x + y = 10 x − y = 6

ⓐ by graphing. ⓑ by substitution. ⓒ Which method do you prefer? Why?

Solve the system of equations { 3 x + y = 12 x = y − 8 { 3 x + y = 12 x = y − 8 by substitution and explain all your steps in words.

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

ⓑ After reviewing this checklist, what will you do to become confident for all objectives?

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## Multi-Step Equations and Inequalities (Algebra 1 - Unit 2) | All Things Algebra®

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## Description

This Multi-Step Equations and Inequalities Unit Bundle contains guided notes, homework assignments, three quizzes, study guide, and a unit test that cover the following topics:

• Multi-Step Equations (Variables on One Side)

• Multi-Step Equations (Variables on Both Sides)

• Special Cases: Infinite Solution & No Solution

• Proportions

• Properties of Equality • Algebraic Proofs

• Literal Equations

• Absolute Value Equations

• Word Problems (Perimeter, Consecutive Integer)

• Multi-Step Inequalities (including interval notation)

• Compound Inequalities (including interval notation)

• Absolute Value Inequalities (including interval notation)

ADDITIONAL COMPONENTS INCLUDED:

(1) Links to Instructional Videos: Links to videos of each lesson in the unit are included. Videos were created by fellow teachers for their students using the guided notes and shared in March 2020 when schools closed with no notice. Please watch through first before sharing with your students. Many teachers still use these in emergency substitute situations. (2) Editable Assessments: Editable versions of each quiz and the unit test are included. PowerPoint is required to edit these files. Individual problems can be changed to create multiple versions of the assessment. The layout of the assessment itself is not editable. If your Equation Editor is incompatible with mine (I use MathType), simply delete my equation and insert your own.

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This resource is included in the following bundle(s):

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More Algebra 1 Units:

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Unit 4 – Linear Equations

Direct & Inverse Variation (Mini-Unit)

Unit 5 – Systems of Equations & Inequalities

Unit 6 – Exponents and Exponential Functions

Unit 7 – Polynomials & Factoring

Unit 8 – Quadratic Equations

Unit 9 – Linear, Quadratic, and Exponential Functions

Unit 10 – Radical Expressions & Equations

Unit 11 – Rational Expressions & Equations

Unit 12 – Statistics

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## Equations and Inequalities Unit 7th Grade TEKS

A 14-day Equations and Inequalities TEKS-Aligned complete unit including writing and solving one-variable two-step equations and inequalities.

## Description

Additional information.

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A 14 day Equations and Inequalities TEKS-Aligned complete unit including: writing and solving one-variable two-step equations and inequalities.

Students will practice with both skill-based problems, real-world application questions, and error analysis to support higher level thinking skills. You can reach your students and teach the standards without all of the prep and stress of creating materials!

Standards: TEKS: 7.10A, 7.10B, 7.10C, 7.11A, 7.11B, 7.11C; Looking for CCSS-Aligned Resources? Grab the Expressions & Equations CCSS-Aligned Unit. Please don’t purchase both as there is overlapping content.

Learning Focus:

- model, write, and solve two-step equations and inequalities
- determine if a value makes an equation true
- represent the solution for equations and inequalities on a number line

## What is included in the 7th grade teks Equations and Inequalities Unit?

1. Unit Overviews

- Streamline planning with unit overviews that include essential questions, big ideas, vertical alignment, vocabulary, and common misconceptions.
- A pacing guide and tips for teaching each topic are included to help you be more efficient in your planning.

2. Student Handouts

- Student-friendly guided notes are scaffolded to support student learning.
- Available as a PDF and the student handouts/homework/study guides have been converted to Google Slides™ for your convenience.

3. Independent Practice

- Daily homework is aligned directly to the student handouts and is versatile for both in class or at home practice.

4. Assessments

- 1-2 quizzes, a unit study guide, and a unit test allow you to easily assess and meet the needs of your students.
- The Unit Test is available as an editable PPT, so that you can modify and adjust questions as needed.

5. Answer Keys

- All answer keys are included.

***Please download a preview to see sample pages and more information.***

How to use this resource:

- Use as a whole group, guided notes setting
- Use in a small group, math workshop setting
- Chunk each student handout to incorporate whole group instruction, small group practice, and independent practice.
- Incorporate our Expressions and Equations Activity Bundle and Inequalities Activity Bundle for hands-on activities as additional and engaging practice opportunities.

Time to Complete:

- Each student handout is designed for a single class period. However, feel free to review the problems and select specific ones to meet your student needs. There are multiple problems to practice the same concepts, so you can adjust as needed.

Is this resource editable?

- The unit test is editable with Microsoft PPT. The remainder of the file is a PDF and not editable.

Looking for more 7th Grade Math Material? Join our All Access Membership Community! You can reach your students without the “I still have to prep for tomorrow” stress, the constant overwhelm of teaching multiple preps, and the hamster wheel demands of creating your own teaching materials.

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## Customer Service

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## This resource is often paired with:

## Inequalities Activity Bundle 7th Grade

## Expressions and Equations Activity Bundle 7th Grade

## Digital Math Activity Bundle 7th Grade

## Expressions and Equations Unit 7th Grade CCSS

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## Unit 2 – Linear Equations and Inequalities

Equations and Their Solutions

LESSON/HOMEWORK

LECCIÓN/TAREA

LESSON VIDEO

EDITABLE LESSON

EDITABLE KEY

SMART NOTEBOOK

Using Inverse Operations to Solve Equations

Linear Equation Solving Review

Justifying Steps in Solving an Equation

Modeling with Linear Equations

Modeling with Linear Equations Involving Integers

Solving Equations with Unspecified Constants

Inequalities

Solving Linear Inequalities

Modeling with Inequalities

Unit Review

Unit 2 Review

UNIT REVIEW

REPASO DE LA UNIDAD

EDITABLE REVIEW

Unit 2 Assessment – Form A

EDITABLE ASSESSMENT

Unit 2 Assessment – Form B

Unit 2 Exit Tickets

Unit 2 Mid-Unit Quiz – Form A

U02.AO.01 – Driving Decisions – Modeling Problem

PDF DOCUMENT - SPANISH

EDITABLE RESOURCE

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## COMMENTS

Unit 2: Equations & Inequalities Homework 13: Inequalities Review ** This is a 2-page document! ** Directions: Solve, graph, and write the solution to each inequality in interval notation. -8X432 -g +5 Interval Notation: C 3 3. 7-(5-4x) < 2(3x+8) I la 2- Interval Notation: 2 < -11 -3x -18 Interval Notation: —q) Of

We develop general methods for solving linear equations using properties of equality and inverse operations. Thorough review is given to review of equation solving from Common Core 8th Grade Math. Solutions to equations and inequalities are defined in terms of making statements true. This theme is emphasized throughout the unit.

Introduction to Systems of Equations and Inequalities; 7.1 Systems of Linear Equations: Two Variables; 7.2 Systems of Linear Equations: Three Variables; 7.3 Systems of Nonlinear Equations and Inequalities: Two Variables; 7.4 Partial Fractions; 7.5 Matrices and Matrix Operations; 7.6 Solving Systems with Gaussian Elimination; 7.7 Solving Systems with Inverses; 7.8 Solving Systems with Cramer's Rule

Learn how to manipulate expressions and solve equations and inequalities. If you're seeing this message, it means we're having trouble loading external resources on our website. ... Unit 2. Rates and percentages. Unit 3. Integers: addition and subtraction. Unit 4. Rational numbers: addition and subtraction. Unit 5. Negative numbers ...

Grade 7 Curriculum Focal Points (NCTM) Number and Operations and Algebra: Developing an understanding of operations on all rational numbers and solving linear equations. Students extend understandings of addition, subtraction, multiplication, and division, together with their properties, to all rational numbers, including negative integers.

n 3 n 2 5 n 2. Rafael forgot to include parentheses around the expression n+2. In his equation, only the n is being multiplied by 3, where he really needs to have the entire n+2 multiplied by 3. Write the correct equation (if you haven't already) and solve it to find the two consecutive even integers.

ultiply a number by 7.Ad. 14 to a number.Subtract. act 6 from a number. vide a number by -3Divide -8 by a. umber.Multiple a number by 4, then ad. 3.Multiply a number by 8, then add -11.I. ubtract 4 from a number, then multiply b. 6.J. Add -1 to a number, then divide by. K.

For every real number a and b, and for c<0, if a>b, the ac<bc; if a<b, then ac>bc. Division Property of Inequality. Compound Inequality. Two inequalities that are joined by the word and or the word or. Study with Quizlet and memorize flashcards containing terms like solution of an inequality, equivalent inequalities, Addition Property of ...

Solving absolute value equations. you must break into seperate cases (case 1 and 2) Solving absolute value equations: Case 1. rewrite problem without absolute value and solve. Solving absolute value equations: Case 2. rewritw without absolute value and negate R side. Last step for solving absolute value equations. plug in to check for solution.

Unit 2: Equations and Inequalities with variables on both sides. Term. 1 / 10. Expression. Click the card to flip 👆. Definition. 1 / 10. A mathematical phrase that contains operations, numbers, and/or variables. Has no equal sign.

The Algebra 1 course, often taught in the 9th grade, covers Linear equations, inequalities, functions, and graphs; Systems of equations and inequalities; Extension of the concept of a function; Exponential models; and Quadratic equations, functions, and graphs. Khan Academy's Algebra 1 course is built to deliver a comprehensive, illuminating, engaging, and Common Core aligned experience!

Unit test. Level up on all the skills in this unit and collect up to 2,200 Mastery points! Start Unit test. Solving equations is a superpower. It means we can model a situation with an equation in any way that makes sense to us, even with an unknown value in the middle. Inequalities are for situations with many true options, like how many pages ...

equation is solved for L. Substitute 2W + 5 for L in the first equation. Solve for W. Substitute W = 13 into the second equation and then solve for L. Step 6. Check the answer in the problem. Does a rectangle with length 31 and width 13 have perimeter 88? Yes. Step 7. Answer the equation. The length is 31 and the width is 13.

Select a Unit. Unit 1 Sequences; Unit 2 Linear and Exponential Functions; Unit 3 Features of Functions; Unit 4 Equations and Inequalities; Unit 5 Systems of Equations and Inequalities; Unit 6 Quadratic Functions; Unit 7 Structures of Quadratic Expressions; Unit 8 More Functions, More Features; Unit 9 Modeling Data

lanaskedelj. Preview. chapter 1. 11 terms. jaeden_micco. Preview. Study with Quizlet and memorize flashcards containing terms like For something to be considered an equation it MUST have ..., What is this considered 3x + 4, Is this an equation, expression, or inequality? 3x+6 > 10 and more.

Description. This Multi-Step Equations and Inequalities Unit Bundle contains guided notes, homework assignments, three quizzes, study guide, and a unit test that cover the following topics: • Multi-Step Equations (Variables on One Side) • Multi-Step Equations (Variables on Both Sides) • Special Cases: Infinite Solution & No Solution.

N-Gen Math Algebra I. The full experience and value of eMATHinstruction courses are achieved when units and lessons are followed in order. Students learn skills in earlier units that they will then build upon later in the course. Lessons can be used in isolation but are most effective when used in conjunction with the other lessons in this course.

Question: Name: Date: Unit 2: Equations & Inequalitles Homework 10: Multh-Step Inequalities Bell:_ Directions: Solve and express the fol lowing inequaities in interval L. 3 interval Notation: Interval Notation: 3.5+8)-7s 23 sx + 33 느 23 3373 Interval Notation: Interval Notation: 6. 2x +5 s 3x- 10 Interval Notation: Interval Notation: 7. 2(x-3)+ 5x s9x-14 Interval

Standards: TEKS: 7.10A, 7.10B, 7.10C, 7.11A, 7.11B, 7.11C; Looking for CCSS-Aligned Resources? Grab the Expressions & Equations CCSS-Aligned Unit. Please don't purchase both as there is overlapping content. Learning Focus: model, write, and solve two-step equations and inequalities; determine if a value makes an equation true

Terms in this set (25) Study with Quizlet and memorize flashcards containing terms like Solve the inequality and graph on a number line. −4x+ 5 ≥ −11, Solve the inequality and graph on a number line. 2.5−3x < −15.5, What is the solution to the equation 4x+7=20 ? and more.

Unit 5 - Systems of Equations & Inequalities (Updated October 2016) copy. Name: Date: Unit 5: Systems of Equations & Inequalities Homework 1: Solving Systems by Graphing ** This is a 2-page document! ** Solve each system of equations by graphing. Clearly identify your solution. -16 — 6y = 30 9x + = 12 +4 v = —12 O Gina Wilson (All Things ...

Algebra 2 -13 - Expressions, Equations, and Inequalities SECTION 1.4: SOLVING EQUATIONS MACC.912.A-CED.A.1.: Create equations and inequalities in one variable and use them to solve problems MACC.912. A-CED.A.4: Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. RATING LEARNING SCALE 4

Lesson 2. Using Inverse Operations to Solve Equations. LESSON/HOMEWORK. LECCIÓN/TAREA. LESSON VIDEO. ANSWER KEY. EDITABLE LESSON. EDITABLE KEY. SMART NOTEBOOK.