GOSEA 6 - Solving Two-Step (& Multi-Step) Equations

GOSA 6

Solving Two-Step (& Multi-Step) Equations

Equations in Geometry will be easier to solve than this one :)

By continuing to the next slide and completing this lesson, you testify that you are the student who is enrolled in GOSA at BCHS with Mrs. Haley and that you are doing the lesson for yourself and no one else.

What is the purpose of solving an equation?

To solve an equation means to find the value of the variable that makes the conditional equation become a true statement when the solution is plugged back into the equation.

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Finding the solution to an equation means you have to isolate the variable (get the variable by itself). There is a process for doing this. You will have to show your work. Trying numbers on your calculator until you find one that works is not "solving an equation".

x + 13 = 25

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Writing down on your paper that 12 + 13 = 25 is not "solving an equation" and will not be accepted as "work".

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To solve this equation, you will have to show the undoing of addition on both sides of the equals sign.

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x + 13 = 25

x + 13 - 13 = 25 - 13

x + 0 = 12

x = 12​

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Not all equations are "one-step" equations.

Solving one-step equations was covered in GOSEA Lesson 5.

One-Step vs Two-Step (& Multi-Step) Equations

A one-step equation is solved in one step:

by undoing addition with subtraction or

by undoing subtraction with addition or

by undoing multiplication with division or

by undoing division with multiplication

One-Step vs Two-Step Equations (continued)

Two-step equations are solved by undoing TWO operations.

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One operation to be undone will be either addition or subtraction.

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The other operation to be undone will be multiplication or division.

The following slide will show you the steps to solving Multi-step (more than two steps) equations.

5 Steps to Solving Equations

(Memorize these steps in order.)

1. Eliminate the parentheses.

2. Add like terms.

3. Eliminate the variable.

4. Eliminate the constant.

5. Eliminate the coefficient.

The next slides will explain what each of the steps means.

Step 1: Eliminate the parentheses.

Not every equation will have parentheses in it.

If an equation does have parentheses in it, then your first task will be to get rid of the parentheses by using the Distributive Property.

The Distributive Property was covered in GOSEA Lesson 3.

Step 2: Add Like Terms

Not every equation will have like terms in it.

If an equation does have like terms, then they (the like terms) will be added AFTER you get rid of the parentheses and BEFORE you start undoing any operations.

Like terms will have to be on the SAME SIDE of an equals sign in order to be added together. There is no work to show in this step.

The addition is done in your head or using a calculator.

Adding like terms was covered in GOSEA Lesson 4.

Step 3: Eliminate the Variable

To be more specific, you will be eliminating the variable term (the term with a variable in it) from one side of the equation.

Every equation you have to solve will have a variable term.

However, you will only need to eliminate the variable term IF there is a variable term on BOTH sides of the equals sign.

Since the variable term will either be added to or subtracted from a constant term, you will eliminate the variable term by either adding or subtracting it (the variable term) from both sides of the equation (Addition/Subtraction Property of Equality - covered in GOSEA Lesson 5)

Step 3 (continued)

2x + 6 = 10

Because there is a variable term on only one side, there is no need to eliminate the variable term.

Step 3 (continued)

3x - 6 = 2x + 8

Because there is a variable term on both sides of the equals sign, it is necessary to eliminate the variable term.

You can eliminate 3x by subtracting 3x from both sides of the equation to get -6 = -1x + 8, or you can eliminate 2x by subtracting 2x from both sides of the equation and getting 1x - 6 = 8.

(I tend to try and make my variable end up on the left side of the equation when possible, but it really does not matter.)

-6 = -1x + 8 and 1x - 6 = 8 are called "equivalent equations" because they have the same solutions, x = 14.

Step 4: Eliminate the Constant

Eliminating a constant term is only necessary if there is a constant term on both sides of the equation (equals sign).

Since the constant term could either be added to or subtracted from a variable term, you will eliminate the constant term by either adding or subtracting it (the constant term) from both sides of the equation (Addition/Subtraction Property of Equality - covered in GOSEA Lesson 5)

Step 4 (continued)

5x = 15

There is only one constant term on one side of the equation, so eliminating a constant term is not necessary.

Step 4 (continued)

x + 10 = 20

By the time you get to Step 4, the variable term will have already been eliminated in Step 3 (if needed) leaving a variable term on only one side of the equation.

There is a constant on both sides of this equation.

While either constant can be eliminated, it makes more sense to eliminate the constant 10 from both sides instead of the 20 since the whole point of solving an equation is to get the variable by itself.

Step 5: Eliminate the Coefficient

In an earlier lesson, you learned that a coefficient is simply another name for a factor (a number being multiplied) and typically refers to a literal (variable/letter) coefficient.

Since a coefficient is a number being multiplied, you will undo multiplication with division.

Multiple Choice

2x + 6 = 14

What should be your first step to solve this equation?

• Eliminate the parentheses

• Add like terms

• Eliminate the variable

• Eliminate the constant

• Eliminate the coefficient

Multiple Choice

2x + 6 = 14

What should be your first step to solve this equation?

• Add 6 to both sides of the equation

• Divide both sides by 2

• Subtract 6 from both sides of the equation

• Subtract 14 from both sides of the equation

Multiple Choice

2x + 6 = 14

Which work is correct?

• 2x + 6 - 14 = 14 - 14

• 2x + 6 - 6 = 14 - 6

• 2x + 6 $$\div$$ 2 = 14 $$\div$$ 2

• 2x + 6 - 6 = 14 + 6

Multiple Choice

14 = 3x + 8

What should be your first step to solve this equation?

• Add like terms

• Eliminate the coefficient

• Eliminate the variable

• Eliminate the parentheses

• Eliminate the constant

Multiple Choice

14 = 3x + 8

What should be your first step to solve this equation?

• Subtract 8 from both sides

• Add 8 to both sides

• Subtract 14 from both sides

• Divide both sides by 3

Multiple Choice

14 = 3x + 8

Which work is correct?

• 14 + 8 = 3x + 8 + 8

• 14 $$\div$$ 3=3x+8 $$\div$$ 3

• 14 - 8 = 3x + 8 - 8

• 14 - 8 = 3x + 8 + 8

Multiple Choice

5 + 2x = 7

What should be your first step to solve this equation?

• Eliminate the parentheses

• Add like terms

• Eliminate the variable

• Eliminate the constant

• Eliminate the coefficient

Multiple Choice

5 + 2x = 7

What should be your first step to solve this equation?

• Subtract 7 from both sides

• Add 5 to both sides

• Subtract 5 from both sides

• Divide both sides by 2

Multiple Choice

5 + 2x = 7

What is the correct work?

• 5+2x $$\div$$ 2 =7 $$\div$$ 2

• 5 + 2x - 7 = 7 - 7

• 5 + 2x + 5 = 7 + 5

• 5 + 2x - 5 = 7 - 5

Multiple Choice

23 = 18 - 2x

What is the first step to solving this equation?

• Add like terms

• Eliminate the ( )

• Eliminate the constant

• Eliminate the variable

• Eliminate the coefficient

Multiple Choice

23 = 18 - 2x

What is the first step to solving this equation?

• Subtract 23 from both sides

• Subtract 18 from both sides

• Subtract 2 from both sides

• Add 18 to both sides

Multiple Choice

23 = 18 - 2x

Which work is correct?

• 23 - 18 = 18 - 2x - 18

• 23 + 18 =18 - 2x + 18

• 23 -23 = 18 - 2x -23

• $$23\div2=18-2x\div2$$

Multiple Choice

What is the first step to solving this equation?

2(x + 6) = 33

• Eliminate the coefficient

• Eliminate the constant

• Eliminate the ( )

• Add like terms

• Eliminate the variable

Multiple Choice

What is the first step to solving this equation?

5(x + 6) - 3x + 15 = 40

• Eliminate the coefficient

• Eliminate the ( )

• Eliminate the constant

• Add like terms

• Eliminate the variable

Multiple Choice

What is the first step to solving this equation?

15 + 23x - 17 = 3x - 5

• Eliminate the coefficient

• Eliminate the ( )

• Eliminate the variable

• Add like terms

• Eliminate the constant

Multiple Choice

What is the next step in solving this equation?

2(x + 5) = 32

2x + 10 = 32

• 2x + 10 - 32 = 32 - 32

• 2x + 10 - 10 = 32 - 10

• 2x + 10 $$\div$$ 2 = 32 $$\div$$ 2

Multiple Choice

What is the next step in solving this equation?

3 + 4(x - 1) + 5 = 2x - 10

3 + 4x - 4 + 5 = 2x - 10

• 3 + 4x - 4 + 5 - 3 + 4 - 5 = 2x - 10 - 3 + 4 - 5

• 4x + 4 = 2x - 10

Multiple Choice

What is the next step in solving this equation?

5x + 13 = 18 - 9x

• 5x + 13 - 13 = 18 - 9x - 13

• 5x + 13 + 9x = 18 - 9x + 9x

Multiple Choice

What is the next step in solving this equation?

4 - 2(x + 5) = 4(x + 2) + 10

4 - 2x - 10 = 4x + 8 + 10

• -2x - 6 = 4x + 18

• 4 - 2x - 10 - 4 + 10 = 4x + 8 + 10 - 4 + 10

Keep doing this assignment until you make a 100. That is the only way you will get the 100 in Geometry in the Fall.​