Lynn Marecek; MaryAnne Anthony-Smith; Andrea Honeycutt Mathis; Jay Abramson; Sharon North; Amy Baldwin; Alyssa Howell

1.7.2 Explaining Acceptable Moves to Solve an Equation

1.7.2 • Explaining Acceptable Moves to Solve an Equation

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 $x$ 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 $x$ 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.	$16 = 4 (9 - x)$ $16 = 36 - 4 x$	$9 x = 5 x + 4$ $14 x = 4$
2.	$5 x = 24 + 2 x$ $3 x = 24$	$\frac{1}{2} x - 8 = 9$ $x - 8 = 18$
3.	$- 3 (2 x + 9) = 12$ $2 x + 9 = - 4$	$6 x - 6 = 3 x$ $x - 1 = 3 x$
4.	$5 x = 3 - x$ $5 x = - x + 3$	$- 11 (x - 2) = 8$ $x - 2 = 8 + 11$
5.	$18 = 3 x - 6 + x$ $18 = 4 x - 6$	$4 - 5 x = 24$ $5 x = 20$

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.

Compare your answer: Your answer may vary, but here is a sample.

Column A: For example:

By the distributive property, $4 (9 - x)$ is equal to $36 - 4 x$ for any value of $x$ . If $4 (9 - x)$ is equal to $16$ , then $36 - 4 x$ is also equal to $16$ .
If the two sides of the equation are equal, then they are still equal if the same amount, in this case $2 x$ , is subtracted from each side.
$- 3 (2 x + 9)$ divided by $- 3$ is $2 x + 9$ and $12$ divided by - $3$ is $- 4$ . Because $- 3 (2 x + 9)$ is equal to $12$ , dividing each of them by $- 3$ gives the same result, so it is true that $2 x + 9$ is equal $- 4$ .
Subtracting by a number is equivalent to adding its opposite, so $3 - x$ is equivalent to $3 + (- x)$ . Changing the order in which we add two numbers doesn't change the sum, per commutative property of addition. If $5 x$ is equal to $3 + - x$ , then it is also equal to $- x + 3$ .
Using the commutative property, $3 x - 6 + x$ can be rewritten as $3 x + x - 6$ . Combining like terms gives an equivalent expression, $4 x - 6$ , so $4 x - 6$ is also equal to $18$ .

2.

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

Compare your answer: Your answer may vary, but here is a sample.

Column B: For example:

If the expressions $9 x$ and $5 x + 4$ are equal, then adding $5 x$ to one expression and subtracting $5 x$ from the other make the two expressions unequal.
The 9 on the right side of the equation is multiplied by 2. If the expression on the left side was also multiplied by 2, the two sides would stay equal, but only a part of the expression on the left is doubled, so the two sides of the equation are no longer equal.
If two expressions are equal, dividing one side by 6 but not dividing the other side by 6 makes them unequal.
The left side of the first equation is divided by $- 11$ but the right side is being added by 11. Performing two different operations with different numbers on the same value may not keep an equation true.
Subtracting 4 from two values that are equal keeps the equation true. The expression on the left was multiplied (or divided) by $- 1$ , however, the expression on the right was not, so the two expressions are no longer equal.

Self Check

Select all of the following equations that are equivalent to

3(x - 2) = 18

.

$3x = 12$
$3x - 6 = 18$
$3x - 2 = 18$
$x - 18 = 6$

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.

$\begin{matrix} - 6 (x + 3) & = & 24 \\ - 6 x - 18 & = & 24 \end{matrix}$

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

Nothing to do. All the variables of $x$ 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.

$\begin{matrix} - 6 x - 18 + 18 & = & 24 + 18 \\ - 6 x & = & 42 \end{matrix}$

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

Use the Multiplication or Division Property of Equality.

$\begin{matrix} \frac{- 6 x}{- 6} & = & \frac{42}{- 6} \\ x & = & - 7 \end{matrix}$

Step 5 - Check the solution.

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

$\begin{matrix} - 6 (x + 3) & = & 24 \\ - 6 (- 7 + 3) & = & 24 \\ - 6 (- 4) & = & 24 \\ 24 & = & 24 ✓ \end{matrix}$

Example 2

Find the solution for $3 (4 x - 1) - 2 = 8 x + 3$ .

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

$\begin{matrix} 3 (4 x - 1) - 2 & = & 8 x + 3 \\ 12 x - 3 - 2 & = & 8 x + 3 \\ 12 x - 5 & = & 8 x + 3 \end{matrix}$

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

$\begin{matrix} 12 x - 5 - 8 x & = & 8 x + 3 - 8 x \\ 4 x - 5 & = & 3 \end{matrix}$

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

$\begin{matrix} 4 x - 5 + 5 & = & 3 + 5 \\ 4 x & = & 8 \end{matrix}$

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

$\begin{matrix} \frac{4 x}{4} & = & \frac{8}{4} \\ x & = & 2 \end{matrix}$

Step 5 - Check the solution.

$\begin{matrix} 3 (4 x - 1) - 2 & = & 8 x + 3 ? \\ 3 (4 (2) - 1) - 2 & = & 8 (2) + 3 ? \\ 3 (8 - 1) - 2 & = & 16 + 3 ? \\ 3 (7) - 2 & = & 19 ? \\ 21 - 2 & = & 19 ? \\ 19 & = & 19 ✓ \end{matrix}$

Example 3

Solve $12 x + 3 (x + 7) = 10 x - 24$ .

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

$\begin{matrix} 12 x + 3 (x + 7) = 10 x - 24 \\ 12 x + 3 x + 21 = 10 x - 24 \\ 15 x + 21 = 10 x - 24 \end{matrix}$

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

$\begin{matrix} 15 x + 21 - 10 x & = & 10 x - 24 - 10 x \\ 5 x + 21 & = & - 24 \end{matrix}$

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

$\begin{matrix} 5 x + 21 - 21 & = & - 24 - 21 \\ 5 x & = & - 45 \end{matrix}$

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

$\begin{matrix} \frac{5 x}{5} & = & - \frac{45}{5} \\ x & = & - 9 \end{matrix}$

Step 5 - Check the solution.

$\begin{matrix} 12 x + 3 (x + 7) & = & 10 x - 24 ? \\ 12 (- 9) + 3 (- 9 + 7) & = & 10 (- 9) - 24 ? \\ - 108 + 3 (- 2) & = & - 90 - 24 ? \\ - 108 + - 6 & = & - 86 ? \\ - 114 & = & - 114 ✓ \end{matrix}$

Video: Acceptable Moves

Watch the following video to learn more about acceptable moves.

Simplifying Equations with Inverse Operations

Try it

Solving Equations and Creating Equivalent Equations

1.

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

Compare your answer:

Your answer may vary, but here are some samples.

$2 x + 8 = 24$
$x + 4 = 6$
$2 x = 16$

2.

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

Compare your answer:

Step 1 - Simplify each side of the equation as much as possible. Use the Distributive Property to remove any parentheses.

$\begin{matrix} 2 (x + 4) & = & 12 \\ 2 x + 8 & = & 12 \end{matrix}$

Step 2 - Collect all the variable terms on one side of the equation. Nothing to do. All the variables of x 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.

$\begin{matrix} 2 x + 8 - 8 & = & 12 - 8 \\ 2 x & = & 4 \end{matrix}$

Step 4 - Make the coefficient of the variable term equal to 1. Use the Multiplication or Division Property of Equality.

$\begin{matrix} \frac{2 x}{2} & = & \frac{4}{2} \\ x & = & 2 \end{matrix}$

Step 5 - Check the solution.

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

$\begin{matrix} 2 (x + 4) & = & 12 \\ 2 (2 + 4) & = & 12 \\ 2 (6) & = & 12 \\ 12 & = & 12 ✓ \end{matrix}$

3.

Solve $- 12 + 8 (x - 5) = - 4 + 3 (5 x - 2)$

Compare your answer:

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

$\begin{matrix} - 12 + 8 (x - 5) & = & - 4 + 3 (5 x - 2) \\ - 12 + 8 x - 40 & = & - 4 + 15 x - 6 \\ 8 x - 52 & = & 15 x - 10 \end{matrix}$

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

$\begin{matrix} 8 x - 52 - 15 x & = & 15 x - 10 - 15 x \\ - 7 x - 52 & = & - 10 \end{matrix}$

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

$\begin{matrix} - 7 x - 52 + 52 & = & - 10 + 52 \\ - 7 x & = & 42 \end{matrix}$

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

$\begin{matrix} \frac{- 7 x}{- 7} & = & \frac{42}{- 7} \\ x & = & - 6 \end{matrix}$

Step 5 - Check the solution.

$\begin{matrix} - 12 + 8 (x - 5) & = & - 4 + 3 (5 x - 2) ? \\ - 12 + 8 (- 6 - 5) & = & - 4 + 3 (5 (- 6) - 2) ? \\ - 12 + 8 (- 11) & = & - 4 + 3 (- 30 - 2) ? \\ - 12 + - 88 & = & - 4 + 3 (- 32) ? \\ - 100 & = & - 4 + - 96 \\ - 100 & = & - 100 ✓ \end{matrix}$