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

1.6.3 Exploring Related Equations

1.6.3 • Exploring Related Equations

Activity

With a partner, discuss the following situations and equations. For each, consider:

What does the solution mean in the context of the situation?
Are the given values solutions?

If operations are applied correctly, the solution to an equation is also the solution to all equations equivalent to it. If operations are incorrectly applied, the solution of each equation is different.

For questions 1 and 2, use the following situation and equation:

Noah is buying a pair of jeans and using a coupon for 10% off. The total price is $56.70, which includes $2.70 in sales tax. Noah’s purchase can be modeled by the equation:

$x - 0.1 x + 2.70 = 56.70$

1.

What does the solution to the equation mean in this situation?

Compare your answer:

Your answer may vary, but here is a sample. The solution is the price of a pair of jeans.

2.

How can you verify that 70 is not a solution but 60 is the solution?

Compare your answer:

Your answer may vary, but here are some samples.

70 can be substituted into the equation to make a false statement: $70 - 0.1 (70) + 2.70 = 56.70$ $65.70 \neq 56.70$ ❌

60 can be substituted into the equation to make a true statement: $60 - 0.1 (60) + 2.70 = 56.70$ $56.70 = 56.70$ $✓$

Here are some equations that are related to $x - 0.1 x + 2.70 = 56.70$ . Each equation is a result of performing one or more moves on that original equation. Each can also be interpreted in terms of Noah’s purchase.

For questions 3 – 5, use the equation $100 x - 10 x + 270 = 5, 670$ .

3.

What was done to Noah’s equation to make this one?

Compare your answer:

Your answer may vary, but here is a sample. Every term was multiplied by 100. The equation is equivalent to Noah’s equation.

4.

What is the interpretation of this new equation?

Compare your answer:

Your answer may vary, but here is a sample. The price is now expressed in cents instead of dollars.

5.

Is the solution the same?

Compare your answer:

Your answer may vary, but here is a sample. 60 is a solution in both equations.

For questions 6 – 8, use the equation $x - 0.1 x = 54$ .

6.

What was done to Noah’s equation to make this one?

Compare your answer:

Your answer may vary, but here is a sample. Subtract 2.70 from both sides of the equation. The equation is equivalent to Noah’s equation.

7.

What is the interpretation of this new equation?

Compare your answer:

Your answer may vary, but here is a sample. The price is still in dollars.

8.

Is the solution the same?

Compare your answer:

Your answer may vary, but here is a sample. 60 is a solution in both equations.

For questions 9 – 11, use the equation $0.9 x + 2.70 = 56.70$ .

9.

What was done to Noah’s equation to make this one?

Compare your answer:

Your answer may vary, but here is a sample. The terms $x$ and 0.1x were correctly combined to create an equivalent equation to Noah’s original equation.

10.

What is the interpretation of this new equation?

Compare your answer:

Your answer may vary, but here is a sample.

10% off means paying 90% of the original price. 90% of the original price plus sales tax is $56.70.

11.

Is the solution the same?

Compare your answer:

Your answer may vary, but here is a sample.

60 is a solution in both equations.

For questions 12 – 14, use the equation $x - 0.1 x = 56.70$ .

12.

What was done to Noah’s equation to make this one?

Your answer may vary, but here is a sample.

Sales tax was not included on the left but was included on the right. This equation is not equivalent to Noah’s original equation.

13.

What is the interpretation of this new equation?

Your answer may vary, but here is a sample.

The price after using the coupon for 10% off before sales tax is not the same as the price after using the coupon for 10% off including the sales tax, $56.70.

14.

Is the solution the same?

Your answer may vary, but here is a sample.

60 is not a solution to this equation.

For questions 15 – 17, use the equation $x - 0.1 x = 59.40$

15.

What was done to Noah’s equation to make this one?

Your answer may vary, but here is a sample. Subtract 2.70 from the left and add 2.70 to the right.

16.

What is the interpretation of this new equation?

Your answer may vary, but here is a sample.

The same operation must be done to both sides of an equation to keep it balanced and to create an equivalent equation. The price after using the coupon for 10% off is not equal to 56.70 plus tax.

17.

Is the solution the same?

Your answer may vary, but here is a sample. 60 is not a solution to this equation.

For questions 18–20, use the equation $2 (x - 0.1 x + 2.70) = 56.70$ .

18.

What was done to Noah’s equation to make this one?

Your answer may vary, but here is a sample. The left side only was multiplied by 2.

19.

What is the interpretation of this new equation?

Your answer may vary, but here is a sample. Two pairs of jeans after the 10% discount and sales tax cost 56.70.

20.

Is the solution the same?

Your answer may vary, but here is a sample. 60 is not a solution to this equation.

21.

Based on your work above, which of the six equations are equivalent to the original equation, $x - 0.1 x + 2.70 = 56.70$ ? Select the three equations that are equivalent to the original.

$100 x - 10 x + 270 = 5670$
$x - 0.1 x = 54$
$0.9 x + 2.70 = 56.70$
$x - 0.1 x = 56.70$
$x - 0.1 x = 59.40$
$2 (x - 0.1 x + 2.70) = 56.70$

Compare your answer: A, B, C were all equivalent equations to the original with 60 as a solution.

$100 x - 10 x + 270 = 5670$ , $x - 0.1 x = 54$ , and $0.9 x + 2.70 = 56.70$ .

Video: Looking at Equivalent Equations

Watch the following video to learn more about why these are equivalent equations.

Looking at Equivalent Equations

Self Check

Which of the following is a correct next step to create an equation equivalent to

x - 0.2x - 4.2 =13.4

?

$2(x - 0.2x) = 17.6$
$x - 0.2x = 9.2$
$10(x - 0.2x-4.2) = 13.4$
$0.8x - 4.2 = 13.4$

Additional Resources

Properties of Equality

When you add, subtract, multiply, or divide the same quantity from both sides of an equation, you still have equality.

Subtraction Property of Equality

For any real numbers $a$ , $b$ , and $c$ , if $a = b$ , then $a - c = b - c$ .

Addition Property of Equality

For any real numbers $a$ , $b$ , and $c$ , if $a = b$ , then $a + c = b + c$ .

Division Property of Equality

For any real numbers $a$ , $b$ , and $c$ , and $c \neq 0$ , if $a = b$ , then $\frac{a}{c} = \frac{b}{c}$ .

Multiplication Property of Equality

For any real numbers $a$ , $b$ , and $c$ , if $a = b$ , then $a c = b c$ .

For equations to be equivalent, inverse operations are used and must be applied to both sides of an equation so it remains balanced.

Inverse operations are operations that “undo” other operations.

Example 1

Solve $3 x = 15$ .

Since 3 is being multiplied by $x$ , to solve for $x$ , divide both sides by 3.

$\frac{3 x}{3} = \frac{15}{3}$

$x = 5$

Example 2

Solve $\frac{x}{2} + 4 = 12$ .

Step 1 - Subtract 4 from both sides.
$\frac{x}{2} + 4 - 4 = 12 - 4$

Step 2 - Simplify.
$\frac{x}{2} = 8$

Step 3 - Multiply both sides by 2.
$2 \times \frac{x}{2} = 8 \times 2$

Step 4 - Simplify.
$x = 16$

Try it

Properties of Equality

For questions 1 – 2, use the scenario:

Denae bought 6 pounds of grapes for $10.74.

1.

Write an equation for the situation.

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

Let $g$ =one pound of grapes.

6g=10.74

2.

Solve the equation.

1.79

Step 1 - Identify the variable.

$g$ : cost for one pound of grapes

Step 2 - Write an equation.

$6 g = 10.74$

Step 3 - Solve for g.

Divide BOTH sides by 6. Divide BOTH sides by 6.

$\frac{6 g}{6} = \frac{10.74}{6}$

Step 4 - Simplify to solve.

Simplify to solve.

$g = 1.79$

The grapes cost $1.79 per pound.