Lesson Objectives

- Demonstrate an understanding of the Addition Property of Equality
- Demonstrate an understanding of the Multiplication Property of Equality
- Demonstrate an understanding of how to check a proposed solution for an Equation
- Learn how to solve Multi-Step Linear Equations

## How to Solve Multi-Step Linear Equations in One Variable

In the previous two lessons, we learned how to solve an equation using
addition and subtraction and how to solve an equation
using multiplication and division.
In this lesson, we will combine our knowledge and learn how to solve any linear equation
in one variable. We must recall from our previous lessons a few key properties:

Example 1: Solve each equation

-5(4x - 4) = -x - 37

Step 1) Simplify each side completely:

On the left, we will use our distributive property to remove parentheses:

-5(4x - 4) » -20x + 20

On the right, we cannot simplify any further. Let's put our two sides together:

-20x + 20 = -x - 37

Step 2) Isolate the variable term:

We will use our addition property of equality to move all variable terms to one side and all numbers to the other:

-20x + 20 = -x - 37

Normally, people like the variable term on the left. It doesn't matter, but we will follow this convention:

Let's add x to each side of the equation. This will remove -x from the right side:

-20x + 20 + x = -x + x - 37

We know that: -x + x = 0

-20x + 20 + x = 0 - 37

0 - 37 = 37, so we can remove the zero:

-20x + 20 + x = -37

On the left side, we have like terms that can be combined: -20x and x

-20x + x » -19x

-19x + 20 = -37

As a last step, we want to get rid of the 20 from the left side of the equation. This will allow the term -19x to be on the left side alone. Let's add (-20) or subtract 20 away from each side:

-19x + 20 - 20 = -37 - 20

-19x = -57

Step 3) Isolate the variable:

We have -19 which is multiplying the variable x. We can undo this multiplication with division. Let's divide each side of the equation by -19: $$\frac{-19}{-19}x = \frac{-57}{-19}$$ $$\require{cancel}\frac{1\cancel{-19}}{\cancel{-19}}x = \frac{3\cancel{-57}}{\cancel{-19}}$$ $$1x = 3$$ Multiplication by 1 leaves a number unchanged. We can show 1x as just x:

$$x = 3$$ Step 4) Check

Let's plug in a 3 for each occurrence of x in our original equation:

-5(4x - 4) = -x - 37

-5(4(3) - 4) = -3 - 37

-5(12 - 4) = -3 - 37

-5(8) = -40

-40 = -40

We can see the same result on each side of the equation (-40). This tells us our solution (x = 3) is correct.

Example 2: Solve each equation

40 - 8x = -3(5x + 3)

Step 1) Simplify each side completely:

We can't simplify anything on the left side of the equation

On the right, we will use our distributive property to remove parentheses:

-3(5x + 3) » -15x - 9

Let's put our two sides together:

40 - 8x = -15x - 9

Step 2) Isolate the variable term:

We will use our addition property of equality to move all variable terms to one side and all numbers to the other:

Let's add 15x to each side of the equation. This will remove -15x from the right side:

40 - 8x + 15x = -15x + 15x - 9

We know that: -15x + 15x = 0

40 - 8x + 15x = 0 - 9

0 - 9 = -9, so we can remove the zero:

40 - 8x + 15x = -9

On the left side, we have like terms that can be combined: -8x and 15x

-8x + 15x » 7x

40 + 7x = -9

As a last step, we want to get rid of the 40 from the left side of the equation. This will allow the term 7x to be on the left side alone. Let's add (-40) or subtract 40 away from each side:

7x + 40 - 40 = -9 - 40

7x = -49

Step 3) Isolate the variable:

We have 7 which is multiplying the variable x. We can undo this multiplication with division. Let's divide each side of the equation by 7: $$\frac{7}{7}x = \frac{-49}{7}$$ $$\require{cancel}\frac{1\cancel{7}}{\cancel{7}}x = \frac{-7\cancel{-49}}{\cancel{7}}$$ $$1x = -7$$ Again, we know that 1x is the same as just x: $$x = -7$$ Step 4) Check

Let's plug in a (-7) for each occurrence of x in our original equation:

40 - 8x = -3(5x + 3)

40 - 8(-7) = -3(5(-7) + 3)

40 + 56 = -3(-35 + 3)

96 = -3(-32)

96 = 96

We can see the same result on each side of the equation (96). This tells us our solution (x = -7) is correct.

Example 3: Solve each equation

-3(1 - x) = 7(x + 6) - 1

Step 1) Simplify each side completely

-3(1 - x) » -3 + 3x

7(x + 6) - 1 » 7x + 42 - 1 » 7x + 41

3x - 3 = 7x + 41

Step 2) Isolate the variable term

3x - 3 - 7x = 7x - 7x + 41

3x - 3 - 7x = 41

-4x - 3 = 41

-4x - 3 + 3 = 41 + 3

-4x = 44

Step 3) Isolate the variable

$$\frac{-4}{-4}x = \frac{44}{-4}$$ $$\frac{1\cancel{-4}}{\cancel{-4}}x = \frac{11\cancel{44}}{-1\cancel{4}}$$ $$x = -11$$ Step 4) Check

-3(1 - x) = 7(x + 6) - 1

-3(1 - (-11)) = 7(-11 + 6) - 1

-3(12) = 7(-5) - 1

-36 = -35 - 1

-36 = -36

We can see the same result on each side of the equation (-36). This tells us our solution (x = -11) is correct.

- Addition Property of Equality - allows us to add/subtract the same value to/from both sides of an equation
- Additive Inverse Property - tells us a number plus its opposite results in 0
- Additive Identity Property - tells us adding zero to a number leaves the number unchanged
- Multiplication Property of Equality - allows us to multiply/divide the same non-zero number by both sides of an equation
- Multiplicative Inverse Property - tells us a number times its reciprocal is 1
- Multiplicative Identity Property - tells us that multiplying a number by 1 leaves the number unchanged
- Distributive Property of Multiplication - tells us we can distribute multiplication over addition or subtraction

### Solving Linear Equations » Four-Step Procedure:

- Simplify each side completely
- Remove any parentheses using the distributive property
- Combine all like terms

- Isolate the variable term (the term that contains the variable) on one side of the equation
- We do this using our addition property of equality - Move all variable terms to one side and all constants (numbers) to the other

- Isolate the variable
- We do this using our multiplication property of equality

- Check the result
- We can check our result by plugging the proposed solution in for each occurrence of the variable in the original equation. If after simplifying, the result is the same value on each side, our solution is correct.

Example 1: Solve each equation

-5(4x - 4) = -x - 37

Step 1) Simplify each side completely:

On the left, we will use our distributive property to remove parentheses:

-5(4x - 4) » -20x + 20

On the right, we cannot simplify any further. Let's put our two sides together:

-20x + 20 = -x - 37

Step 2) Isolate the variable term:

We will use our addition property of equality to move all variable terms to one side and all numbers to the other:

-20x + 20 = -x - 37

Normally, people like the variable term on the left. It doesn't matter, but we will follow this convention:

Let's add x to each side of the equation. This will remove -x from the right side:

-20x + 20 + x = -x + x - 37

We know that: -x + x = 0

-20x + 20 + x = 0 - 37

0 - 37 = 37, so we can remove the zero:

-20x + 20 + x = -37

On the left side, we have like terms that can be combined: -20x and x

-20x + x » -19x

-19x + 20 = -37

As a last step, we want to get rid of the 20 from the left side of the equation. This will allow the term -19x to be on the left side alone. Let's add (-20) or subtract 20 away from each side:

-19x + 20 - 20 = -37 - 20

-19x = -57

Step 3) Isolate the variable:

We have -19 which is multiplying the variable x. We can undo this multiplication with division. Let's divide each side of the equation by -19: $$\frac{-19}{-19}x = \frac{-57}{-19}$$ $$\require{cancel}\frac{1\cancel{-19}}{\cancel{-19}}x = \frac{3\cancel{-57}}{\cancel{-19}}$$ $$1x = 3$$ Multiplication by 1 leaves a number unchanged. We can show 1x as just x:

$$x = 3$$ Step 4) Check

Let's plug in a 3 for each occurrence of x in our original equation:

-5(4x - 4) = -x - 37

-5(4(3) - 4) = -3 - 37

-5(12 - 4) = -3 - 37

-5(8) = -40

-40 = -40

We can see the same result on each side of the equation (-40). This tells us our solution (x = 3) is correct.

Example 2: Solve each equation

40 - 8x = -3(5x + 3)

Step 1) Simplify each side completely:

We can't simplify anything on the left side of the equation

On the right, we will use our distributive property to remove parentheses:

-3(5x + 3) » -15x - 9

Let's put our two sides together:

40 - 8x = -15x - 9

Step 2) Isolate the variable term:

We will use our addition property of equality to move all variable terms to one side and all numbers to the other:

Let's add 15x to each side of the equation. This will remove -15x from the right side:

40 - 8x + 15x = -15x + 15x - 9

We know that: -15x + 15x = 0

40 - 8x + 15x = 0 - 9

0 - 9 = -9, so we can remove the zero:

40 - 8x + 15x = -9

On the left side, we have like terms that can be combined: -8x and 15x

-8x + 15x » 7x

40 + 7x = -9

As a last step, we want to get rid of the 40 from the left side of the equation. This will allow the term 7x to be on the left side alone. Let's add (-40) or subtract 40 away from each side:

7x + 40 - 40 = -9 - 40

7x = -49

Step 3) Isolate the variable:

We have 7 which is multiplying the variable x. We can undo this multiplication with division. Let's divide each side of the equation by 7: $$\frac{7}{7}x = \frac{-49}{7}$$ $$\require{cancel}\frac{1\cancel{7}}{\cancel{7}}x = \frac{-7\cancel{-49}}{\cancel{7}}$$ $$1x = -7$$ Again, we know that 1x is the same as just x: $$x = -7$$ Step 4) Check

Let's plug in a (-7) for each occurrence of x in our original equation:

40 - 8x = -3(5x + 3)

40 - 8(-7) = -3(5(-7) + 3)

40 + 56 = -3(-35 + 3)

96 = -3(-32)

96 = 96

We can see the same result on each side of the equation (96). This tells us our solution (x = -7) is correct.

Example 3: Solve each equation

-3(1 - x) = 7(x + 6) - 1

Step 1) Simplify each side completely

-3(1 - x) » -3 + 3x

7(x + 6) - 1 » 7x + 42 - 1 » 7x + 41

3x - 3 = 7x + 41

Step 2) Isolate the variable term

3x - 3 - 7x = 7x - 7x + 41

3x - 3 - 7x = 41

-4x - 3 = 41

-4x - 3 + 3 = 41 + 3

-4x = 44

Step 3) Isolate the variable

$$\frac{-4}{-4}x = \frac{44}{-4}$$ $$\frac{1\cancel{-4}}{\cancel{-4}}x = \frac{11\cancel{44}}{-1\cancel{4}}$$ $$x = -11$$ Step 4) Check

-3(1 - x) = 7(x + 6) - 1

-3(1 - (-11)) = 7(-11 + 6) - 1

-3(12) = 7(-5) - 1

-36 = -35 - 1

-36 = -36

We can see the same result on each side of the equation (-36). This tells us our solution (x = -11) is correct.

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