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Algebra 1

1.6.3 Exploring Related Equations

Algebra 11.6.3 Exploring Related Equations

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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 numbers 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:

x0.1x+2.70=56.70x0.1x+2.70=56.70

1.

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

2.

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

3.

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

4.

What is the interpretation of this new equation?

5.

Is the solution the same?

6.

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

7.

What is the interpretation of this new equation?

8.

Is the solution the same?

9.

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

10.

What is the interpretation of this new equation?

11.

Is the solution the same?

For numbers 12 – 14, use the equation x0.1x=56.70x0.1x=56.70.

12.

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

13.

What is the interpretation of this new equation?

14.

Is the solution the same?

For numbers 15–17, use the equation x0.1x=59.40x0.1x=59.40

15.

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

16.

What is the interpretation of this new equation?

17.

Is the solution the same?

For numbers 18–20, use the equation 2(x0.1x+2.70)=56.702(x0.1x+2.70)=56.70.

18.

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

19.

What is the interpretation of this new equation?

20.

Is the solution the same?

21.

Based on your work above, which of the six equations are equivalent to the original equation, x0.1x+2.70=56.70x0.1x+2.70=56.70? Select the three equations that are equivalent to the original.

  • 100 x 10 x + 270 = 5670 100x10x+270=5670
  • x 0.1 x = 54 x0.1x=54
  • 0.9 x + 2.70 = 56.70 0.9x+2.70=56.70
  • x 0.1 x = 56.70 x0.1x=56.70
  • x 0.1 x = 59.40 x0.1x=59.40
  • 2 ( x 0.1 x + 2.70 ) = 56.70 2(x0.1x+2.70)=56.70

Video: Looking at Equivalent Equations

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

Self Check

Which of the following is a correct next step to create an equation equivalent to x 0.2 x 4.2 = 13.4 ?
  1. 2 ( x 0.2 x ) = 17.6
  2. x 0.2 x = 9.2
  3. 10 ( x 0.2 x 4.2 ) = 13.4
  4. 0.8 x 4.2 = 13.4

Additional Resources

Properties of Equality

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 aa, bb, and cc,if a=ba=b,then ac=bcac=bc.

Addition Property of Equality

For any real numbers aa, bb, and cc,if a=ba=b,then a+c=b+ca+c=b+c.

Division Property of Equality

For any real numbers aa, bb, and cc, and c0c0,if a=ba=b,then ac=bcac=bc.

Multiplication Property of Equality

For any real numbers aa, bb, and cc,if a=ba=b,then ac=bcac=bc.

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 3x=153x=15.

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

3x3=1533x3=153

x=5x=5

Example 2

Solve x2+4=12x2+4=12.

Step 1 - Subtract 4 from both sides.
x2+44=124x2+44=124

Step 2 - Simplify.
x2=8x2=8

Step 3 - Multiply both sides by 2.
2×x2=8×22×x2=8×2

Step 4 - Simplify.
x=16x=16

Try it

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.

2.

Solve the equation.

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