Topic 8 300 Lesson 8-3

1

1

1

1

1

1

1

1

1

1

1

1

1

x

x

x

1

1

1

1

1

1

1

1

1

1

1

1

1

x

x

x

1

1

–1

–1

1

1

1

1

1

1

1

1

1

1

1

–1

–1

x

x

x

1

1

–1

–1

1

1

1

1

1

1

1

1

1

1

1

–1

–1

x

x

x

1

1

1

1

1

1

1

1

1

x

x

x

1

1

1

1

1

1

1

1

1

x

x

x

See your complete lesson at MyMathUniverse.com See your complete lesson at MyMathUniverse.com

Common Core State Standards 6.RP.3, 6.RP.3a

Part 1

VocabularyX-coordinate,Y-coordinate, X-axis, Y-axis, quadrant, origin, ordered pair, coordinate plane

Example Solve an Equation Using Algebra Tiles

Write the modeled equation. Then solve it.

Step 1 Represent equation in algebra

tiles. 3x + 2 = 11

Step 2 Add two –1 tiles to each side.

Step 3 On each side, the –1 and +1 tiles

make two groups of zero pairs.

Solution

Step 4 Remove the zero pairs so there

are three x tiles on the left and

nine +1 tiles on the right.

Step 5 Each x tile is equal to three +1

tiles.

Check

Substitute the answer, x = 3, into the original equation. 3x + 2 = 11

3(3) + 2 ≟ 11

9 + 2 ≟ 11

11 = 11 ✓

Using Scientific Notation to Describe Very Large Quantities8-3

Topic 8 301 Lesson 8-3 See your complete lesson at MyMathUniverse.com

Key ConceptSolving a two-step equation requires transforming the equation into a series

of simpler equivalent equations until you isolate the variable on one side

and a number on the other side.

Solve x + 1.5 = 2.75

Step 1 Undo addition or subtraction

For many equations, you can undo addition or subtraction first.

Solve x + 1.5 = 2.75.

x + 1.5 = 2.75

x + 1.5 - 1.5 = 2.75 - 1.5

x = 1.25

Step 2 Undo multiplication or division

To isolate the variable, undo multiplication or division next. The

coefficient of the variable should be 1.

x = 1.25

(3) Q x R = (3)(1.25)

x = 3.75

Step 3 Check your answer by substituting it into the original equation.

x + 1.5 = 2.75

3.75 + 1.5 ≟ 2.75

1.25 + 1.5 ≟ 2.75

2.75 = 2.75 ✓

Undo addition by subtracting 1.5 from each side.

Undo division by multiplying each side by 3.

Topic 8 302 Lesson 8-3 See your complete lesson at MyMathUniverse.com

Part 2Example Solve an Equation When the Solution Pieces

are Given

Solve the equation 5b - 3.75 = 26.25. Justify each step in solving the

equation.

Write the original equation. 5b - 3.75 = 26.25

Undo subtraction using the 5b - 3.75 + 3.75 = 26.25 + 3.75

Addition Property of Equation.

Simplify. 5b = 30

Undo multiplication using the 5b

=

30

Division Property of Equation.

Simplify. b = 6

5b - 3.75 = 26.25

5(6) - 3.75 ≟ 26.25

30 - 3.75 ≟ 26.25

26.25 ≟ 26.25 ✓

Solution

Check

Substitute 6 for b in the original equation to check your answer.

Topic 8 303 Lesson 8-3

to

Words total cost = cost per ticket

• number of tickets

+ processing fee

Let n = the number of tickets ordered.

212.50 = 39.95 • n + 12.75Equations

See your complete lesson at MyMathUniverse.com

Part 3Example Write and Solve an Equation for a

Real-World Situation

You are ordering concert tickets online. Each ticket costs $39.95 and you

have a processing fee of $12.75. The total cost is $212.50. Write and solve an

equation to find the number of tickets ordered.

Solution

The equation is 212.50 = 39.95 + 12.75.

Now solve the equation to find the number of tickets ordered for the

concert.

Write the original equation. 212.5 = 39.95n + 12.75

Undo subtraction using the 212.5 - 12.75 = 39.95n + 12.75 - 12.75

Addition Property of Equation.

Simplify. 199.75 = 39.95n

Undo multiplication using the 199.75

=

39.95n

Division Property of Equation.

Simplify. 5 = n

You ordered 5 tickets.

Topic 8 304 Lesson 8-3 See your complete lesson at MyMathUniverse.com

Homework

1. Use the algebra tiles to help you solve

the equation 4x + 5 = 13.

2. Use the algebra tiles to help you solve

the equation 3x - 5 = 7.

3. Complete the steps to solve the

equation 6x + 16 = 58.

6y + 16 = 58 Write the original equation.

6y = ■ Apply the Subtraction Property of Equality.

y = ■ Apply the Division Property of Equality.

4. Think About the Process What properties of equality do you need to use to solve the equation 4.9x - 1.9 = 27.5?

5. Write the first step in solving the equation 4m - 12 = 0.

6. Write the first step in solving the equation 3 = 7y + 9.

7. Solve 7x - 7 = 56.

8. Solve 6r - 5 = 19.

9. Solve 125 + 3b = 154.97.

10. Solve 1.50 + = 2.75.

11. Solve -7 = 1 + 2g.

12. When three times a number is decreased by 4, the result is 14.

a. Define a variable to represent the number.

b. Write an equation to represent the situation.

c. Solve the equation to find the number.

13. In 2000, the number of federal hazardous waste sites in State X was 8 less than twice the number of sites in State Y. Suppose there were 34 such sites in State X. Write and solve an equation that represents the number of hazardous waste sites, n, there were in State Y.

14. Writing Solve the equation

3x + 2 = 17 using algebra tiles. Describe a real-world situation that could be modeled with the given equation and algebra tiles.

15. Error Analysis A student solved the

equation 2x + 4 = 10 using algebra tiles.

She incorrectly says that the solution is 7. Solve the equation. What mistake might the student have made?

16. While shopping for clothes, Tracy spent $38 less than 3 times what Darnell spent. Tracy spent $10. Write and solve an equation to find out how much Darnell spent.

Digital Resources8-3

Topic 8 305 Lesson 8-3 See your complete lesson at MyMathUniverse.com

Additional Practice 17. Mental Math Solve n

10 + 7 = 10.

18. Multiple Representations A group of four friends went to the movies. In addition to their tickets, they bought a large bag of popcorn to share for $6.25. The total was $44.25. Write and solve an question to find the cost of one movie ticket.

19. Reasoning Explain why 23 is NOT a

solution for the equation 5 = 9k - 4.

20. Error Analysis Which student’s work is correct? Explain.

Challenge 21. A number n times 26, decreased by

126, is 238. A number m times 9, added to 112, is 265.

a. Write an equation to represent

each situation.

b. Solve the equations for each

variable.

c. Compare the values of m and n.

22. At a party, the number of people who ate salad was 11 less than of the total number of people at the party. The number of people who ate salad was 5. Write and solve an equation to find the total number of people at the party.

23. Consider the equation 0.5x + 1.3 = 4.8.

a. Solve the equation.

b. Multiply each side of the original equation by 10. Write the new equation.

c. Solve the equation from part b for x.

d. How do the solutions for parts a and c compare?

e. Explain why it is helpful to multiply by 10.

24. Writing Write a one-step equation

and a two-step equation such that both equations have a solution of 1.78.

x2

x2

x2

Having trouble with this Practice?

Use this chart to find the example that will help you solve the problem

Exercises See

1, 2, 14, 15 Part 1 Example

3 - 11 Part 2 Example

12, 13, 16 Part3 Example

x2

x2