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A16 Appendix B Exploring Equations and Functions
Solving Equations with
Variables on Both Sides
B.
STATE
STANDARDS
MA.7.A.3. MA.7.A.3. MA.7.A.3. MA.7.G.4.
S
How can you solve an equation that has variables on both sides?
Work with a partner. You have an email pal in Antarctica. Your email pal tells you the temperature in McMurdo. You ask whether he gave the temperature in Celsius or Fahrenheit. He says “It’s the same on both scales.” What is the temperature? a. TABLE Use “Guess, Check, and Revise” with a table to find the only temperature that is the same on both scales.
F C
b. GRAPH Draw the line given by C = F in the coordinate plane. Locate the point at which the graph of C = F intersects the graph of
C = 5 — 9
( F − 32).
c. ALGEBRA Let x be the temperature that is the same on both scales. Substitute x for C and F in the equation
C = 5 — 9
( F − 32).
Then solve for x.
d. Compare your solutions from parts (a)–(c). Did you get the same solution with each method? Which method do you prefer? Why?
11 ACTIVITY: Using a Table, Graph, and Algebra
Antarctic Peninsula
Atlantic Ocean
South Pacific Ocean
Indian Ocean
South Pole
McMurdo
F
C
− 40
− 80
− 120
40
80
120
− 120 − 80 − 40 40 80 120
C = 59 ( F − 32)
Section B.2 Solving Equations with Variables on Both Sides A
Play with a partner. ● Write each expression on a scrap of brown or blue paper. Place the brown pieces of paper in one bag and the blue pieces of paper in another bag. ● Draw an expression from each bag and set them equal to each other. ● If you can solve the equation, you move one space. If you cannot solve the equation, your partner gets a chance to solve it and move one space. ● Put the expressions back into their bags. ● Take turns. The fi rst person to reach the South Pole wins.
22 GAME: Race to the South Pole
Brown Papers Blue Papers x 2 x x + 1 2 x + 4 x − 1 − 2 x x + 2 − 2 x + 4 x − 2 3 x x + 3 3 x + 6 x − 3 − 3 x − 3 x + 6
x + 1 =^3 x
3. IN YOUR OWN WORDS How can you solve an equation that has variables on both sides? Give an example and solve it.
Use what you learned about solving equations with variables on both sides to complete Exercises 4–9 on page A20.
0 200 400 600 800 1000 km
R O S S S E A
ANTARCTICA
Kapp Evans
Framheim Bay of Whales
A M U N D S E N S E A
King Edward VII Land
Marie Byrd Land
Queen Maud Land
George V Land
Amundsen
Amundsen 12/14/1911 Scott 1/17/
Scott
R OR
B W
N DDD SD S SS (^) E NE N S ES EN S ES EEE AE AA Player 1 starts here.
Player 2 starts here.
Section B.2 Solving Equations with Variables on Both Sides A
EXAMPLE 33 Standardized Test Practice
The polygons have the same perimeter. What is the area of the rectangle? A 12 in. B 8 in.^2 C 12 in. 2 D 24 in.^2
The perimeter of the rectangle is x + x + 4 + 4 = 2 x + 8. The perimeter of the triangle is 4 + 2 x + x + 2 = 3 x + 6.
2 x + 8 = 3 x + 6 Write an equation. 8 = x + 6 Subtract 2 x from each side. 2 = x Subtract 6 from each side.
So, the area of the rectangle is 2(4) = 8 square inches. The correct answer is B.
Perimeter of rectangle Perimeter of triangle
EXAMPLE 44 Real-Life Application
A boat travels x miles per hour from Miami to Nassau. The boat travels 3 miles per hour slower on the return trip. What is the distance from Miami to Nassau?
The speed, or rate, of the boat on the return trip is ( x − 3) miles per hour.
10 x = 12( x − 3) Write an equation. 10 x = 12 x − 36 Use Distributive Property. − 2 x = − 36 Subtract 12 x from each side. x = 18 Divide each side by −2.
The boat travels 18 miles per hour from Miami to Nassau in 10 hours. So, the boat travels 18(10) = 180 miles.
The distance from Miami to Nassau is 180 miles.
5. In Example 3, what is the perimeter of the triangle? 6. In Example 4, what is the speed of the boat on the return trip?
Distance from Nassau to Miami
Distance from Miami to Nassau
Exercises 25–
4 in.
4 in.
2 x
x + 2
x
Nassau
Miami
The Bahamas
10 hours
12 hours
B.2 Exercises
A20 Appendix B Exploring Equations and Functions
1. REASONING Describe the steps you would use to solve 5 x = 3 x + 4. 2. OPEN-ENDED Write an equation with variables on both sides that has a solution of −1. 3. WRITING To solve − 3 x + 4 = − 2 x , is it easier to add 3 x to each side or add 2 x to each side? Explain.
9 +(-6)=3 3 +(-3)= 4 +(-9)= 9 +(-1)=
Solve the equation. Check your solution.
4. 2 x = − x − 6 5. − 9 + c = 4 c 6. − 3 y = 7 y 7. − 5 a = 2 − a 8. 6 w − 5 = 8 w 9. 3 z = 14 + 10 z 10. 4 f + 8 = 9 f − 12 11. 3 p − 11 = 5 p + 6 12. − 7 k + 10 = 9 k + 18 13. − 2 + 2 d = 6 d + 6 14. 8 b − 7 = − 11 + 3 b 15. 12 h − 7 = 6 h + 8 16. ERROR ANALYSIS Describe and correct the error in solving the equation.
Solve the equation. Check your solution.
17. 2 k − 8 = 4( k + 1) 18. 3( g − 3) = 2(6 − 2 g ) 19. −5( f + 7) = 3(3 f − 1) 20. 9.2 − 4 w = −2(3 w + 5) 21. 2.5(3 b − 4) = 3.5 b − 6 b 22. 6(1.5 h − 1) = 5(2.2 h + 3) 23. ERROR ANALYSIS Describe and correct the error in solving the equation. 24. MUSIC LESSONS It costs $50 to be a member of a music club. A member of the club pays $10 per music lesson. A nonmember pays $20 per music lesson. How many music lessons must a member and a nonmember take so that the cost for each is the same?
Help with Homework
2( v − 5) = − (3 v + 5) 2 v − 10 = − 3 v + 5 5 v = 15 v = 3
3 x − 7 = − 2 x + 8 3 x + ( − 2x) = 8 + 7 x = 15
