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Algebra 1

1.7.2 Explaining Acceptable Moves to Solve an Equation

Algebra 11.7.2 Explaining Acceptable Moves to Solve an Equation

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Activity

Here are some pairs of equations. While one partner listens, the other partner should:

  • Choose a pair of equations from column A. Explain why, if xx is a number that makes the first equation true, then it also makes the second equation true.
  • Choose a pair of equations from column B. Explain why the second equation is no longer true for a value of xx that makes the first equation true.

Then, switch roles until you run out of time or you run out of pairs of equations.

Some principles you might use to complete this activity include the following:

  • Distributive Property
  • Commutative Property
  • Addition Property of Equality
  • Additive Inverse
  • Subtraction Property of Equality
  • Multiplication Property of Equality
  • Division Property of Equality
A B
1. 16=4(9x)16=4(9x)16=364x16=364x 9x=5x+49x=5x+414x=414x=4
2. 5x=24+2x5x=24+2x3x=243x=24 12x8=912x8=9x8=18x8=18
3. 3(2x+9)=123(2x+9)=122x+9=42x+9=4 6x6=3x6x6=3xx1=3xx1=3x
4. 5x=3x5x=3x5x=x+35x=x+3 11(x2)=811(x2)=8x2=8+11x2=8+11
5. 18=3x6+x18=3x6+x18=4x618=4x6 45x=2445x=245x=205x=20

After you have discussed these equations with your partner, answer questions 1 - 2.

1.

Which pair of equations did you select from Column A? Explain your reasoning.

2.

Which pair of equations did you select from Column B? Explain your reasoning.

Self Check

Select all of the following equations that are equivalent to 3 ( x 2 ) = 18 .
  1. 3 x = 12
  2. 3 x 6 = 18
  3. 3 x 2 = 18
  4. x 18 = 6

Additional Resources

Solving Equations and Creating Equivalent Equations

One way to create equivalent equations is to solve the equation. In each step of solving, an equivalent equation is created.

Here is a general strategy for solving linear equations:

Step 1 - Simplify each side of the equation as much as possible.

Use the Distributive Property to remove any parentheses. Combine like terms.

Step 2 - Collect all the variable terms on one side of the equation.

Use the Addition or Subtraction Property of Equality.

Step 3 - Collect all the constant terms on the other side of the equation.

Use the Addition or Subtraction Property of Equality.

Step 4 - Make the coefficient of the variable term equal to 1.

Use the Multiplication or Division Property of Equality.

State the solution to the equation.

Step 5 - Check the solution.

Substitute the solution into the original equation to make sure the result is a true statement.

Example 1

Here is an example of solving equations. Notice that each step is an acceptable move that creates an equivalent equation.

Step 1 - Simplify each side of the equation as much as possible.

Use the Distributive Property to remove any parentheses.

6(x+3)=246x18=246(x+3)=246x18=24

Step 2 - Collect all the variable terms on one side of the equation.

Nothing to do. All the variables of xx are on the left side.

Step 3 - Collect all the constant terms on the other side of the equation.

Use the Addition or Subtraction Property of Equality.

6x18+18=24+186x=426x18+18=24+186x=42

Step 4 - Make the coefficient of the variable term equal to 1.

Use the Multiplication or Division Property of Equality.

6x6=426x=76x6=426x=7

Step 5 - Check the solution.

Substitute the solution into the original equation to make sure the result is a true statement.

6(x+3)=246(7+3)=246(4)=2424=246(x+3)=246(7+3)=246(4)=2424=24

Example 2

Find the solution for 3(4x1)2=8x+33(4x1)2=8x+3.

Step 1 - Simplify each side of the equation as much as possible.

3(4x1)2=8x+312x32=8x+312x5=8x+33(4x1)2=8x+312x32=8x+312x5=8x+3

Step 2 - Collect all the variable terms on one side of the equation.

12x58x=8x+38x4x5=312x58x=8x+38x4x5=3

Step 3 - Collect all the constant terms on the other side of the equation.

4x5+5=3+54x=84x5+5=3+54x=8

Step 4 - Make the coefficient of the variable term equal to 1.

4x4=84x=24x4=84x=2

Step 5 - Check the solution.

3(4x1)2=8x+3?3(4(2)1)2=8(2)+3?3(81)2=16+3?3(7)2=19?212=19?19=193(4x1)2=8x+3?3(4(2)1)2=8(2)+3?3(81)2=16+3?3(7)2=19?212=19?19=19

Example 3

Solve 12x+3(x+7)=10x2412x+3(x+7)=10x24.

Step 1 - Simplify each side of the equation as much as possible.

12x+3(x+7)=10x2412x+3x+21=10x2415x+21=10x2412x+3(x+7)=10x2412x+3x+21=10x2415x+21=10x24

Step 2 - Collect all the variable terms on one side of the equation.

15x+2110x=10x2410x5x+21=2415x+2110x=10x2410x5x+21=24

Step 3 - Collect all the constant terms on the other side of the equation.

5x+2121=24215x=455x+2121=24215x=45

Step 4 - Make the coefficient of the variable term equal to 1.

5x5=455x=95x5=455x=9

Step 5 - Check the solution.

12x+3(x+7)=10x24?12(9)+3(9+7)=10(9)24?108+3(2)=9024?108+6=86?114=11412x+3(x+7)=10x24?12(9)+3(9+7)=10(9)24?108+3(2)=9024?108+6=86?114=114

Video: Acceptable Moves

Watch the following video to learn more about acceptable moves.

Try it

Try It: Solving Equations and Creating Equivalent Equations

1.

Write two equivalent equations to 2(x+4)=122(x+4)=12.

2.

Solve 2(x+4)=122(x+4)=12.

3.

Solve 12+8(x5)=4+3(5x2)12+8(x5)=4+3(5x2)

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