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A ry_?:2
y —x + 2x — y < 4 —y < —x + 4 y > x — 4
2.5)Hy>2 y —5x + 2
2( + y 1 y —2x + 1
X
EXAMPLE Solve.0 0
2x + yLs 4 3x -4- 2y_. 6
Answer
x?_ 0
2x+y54
y -2x 4
3x+ 2y s_ 6 2y -3x + 6 k.
3x +3
The solution is all points in the darkest shaded region.
Or* A. Exercises Graph each system of inequalities.
4. 2x — y < 5 —y < —2x + 5 y> 2x — 5
3x + y 4 y —3x + 4 6. 3x + y> 9
y> —3x+ 9
2x y 6 y —2x + 6
3. 3x — 2y 6 x + y 3 —2y —3x + 6 y —x + 3
y 3 — 3
1. x+y_2
x— y < 4
2. 5x+y>2
2x + y5_ 1
6. 3x + y> 9
2x + y 6
7. x+2y_ ._ —4
x+ 2y < 6
3. 3x — 2y 6 x+ y 3
4. 2x — y< 5 3x + y 4
9. x
y 0
x + 4y 8
2x+y4
► B. Exercises Graph each system of inequalities.
5. x+ 5y < 15 3x + 2y_. 8
11. 0
yS2
x+y>1
10. 0
8. 0
y>-0
Y" 0
3x + y 8
5x+ 6y 30
x+2y . 6
5. x + 5y < 15 3x + 2y -5 8 5y < —x+ 15 2y —3x + 8 y < 51 x + 3
Y 23
x + 4
316 CHAPTER 7 SYSTEMS OF EQUATIONS AND INEQUALITIES
316 CHAPTER 7 SYSTEMS OF EQUATIONS AND INEQUALITIES
CLIrrluiative Review16. 2x + y 4
y 5. -2x + 4 x+y2
Find the equation of each line if - x + 2 18. the slope is - 3 and the y- intercept is (0, 2). = -3x + 2 x>_ 0 19. the slope is 3 and it passes through (2, 5). = 3 x + [6.7] y>_0
20. it passes through (1, 5) and ( - 2, 2). = ± 4'6.31
21. it is vertical and passes through ( - 2, - 5). = - _15.3]
22. it is parallel to y = z x - 4 and passes through (1, 5). y = + [6.7]
Is eact. point a solution to the system of inequalities? Do not graph. 12. 5x + 3y 2 x - 2y 4 2 1 - 2(-2) 4
5. x -F y= 11 y= -x + 11 2x - y = - 5 -y = -2x - 5 y 2x + 5
6. 3x - y = 14 x + 3y = -12 = -3x 14
3y = -x - 12
y 3x - 14 y= 31 x 4
Chapter Review
Determine whether ( - 2, 3) is a solution to each system. 1. 3x + 4y = 6
3. x 2y = 4
y = 1 - x
3x + y = - 3 2. 5x - 2y = 7
4. x + y < 2 x + y = 1 x 2y 4
Use the graphing method to solve each system.
5. x+y= 11 7. x + y = - 2
2x - y = -5 3x + y = -10 . , - 4, 2 6. 3x - y = 14 8. x + 3y = 5
x + 3y = -12 y = 2 -
Use the substitution method to solve each system.
9. 2x + 3y = 8
11. 4x + 3y = -20
x + 2y -= 4
5x - 2y = -2
10. 5x - 3y = 12
12. 5x + y = 8
3x + y = -2 -10x - 2y = -3 nc
Use the addition method to solve each system.
13. x - 2y = -11 15. 5x - 3y = -25
3x + y = 16 6x + 2y = - 2 14. 2x + 3y =- 9 16. x + y = -2
4x - y 2 [ I 2x + 2y = -4 en:i!'E,
Solve each system using any method.
17. y = 2x - 3 19. 2x + 5y = 8 y = -g-x + 7,1, x - 5y = 4
18. x - 2y = - 42 20. 0.5x - 2.1y = 7.2
3x - y = - 31 "-4, 1.4x + 0.3y = 7.8
Solve each system using any method. Tell whether each is consistent or inconsistent, dependent or independent. 21. x + y = 5 nc solutior..!
x = 2 - y incorisiste!-.. 22. 3x + 5y = 1 ;2, -1); cc:-.:L•
4x - 3y = 11 ndepende:. 23. 3x + y = 6
2 -3-y = 5 - 2x incons::- .-
320 CHAPTER 7 SYSTEMS OF EQUATIONS AND INEQUALITIES
Tips •
Have three students solve the following system at the board. each using a different method (graphing, substitution, and addition). You may want to have three other students explain their work.
3x - y = 3 (2, 3) 2x + y = 7 Be sure the students who are not at the
board also work the problem on paper for practice. You may want to have a student graph a system of inequalities at this time as well.
Common Student Error. Students often confuse types of word problems. Help them by discussing characteristics and then prac-tice determining the type on several prob-lems. Be sure to include several of each type.
24. 3x + y = 4 ertire 2y = 8 - 6x 7-,7--Zr.rident
25. x + y2 = 8 4 2); x = y2 consistent,. independent
Ex. 28. Requires I = Prt, which results in the answer / = 0.07d(1). Students may leavt the last factor in their answer. n
320 CHAPTER 7 SYSTEMS OF EQUATIONS AND INEQUALITIES
Give a symbolic expression for each. 26. The distance traveled by a jet in 12 hours at x mph 27. The upstream speed of a kayaker that goes 3 mph in still water if
paddling in a current of c mph t - : 28. The annual interest received from an investment of d dollars at 7% = 29. The amount of salt when m gallons of a 20%-salt solution is mixed with
n I gallons of a 23%-salt solution 0.2m +
Solve algebraically. 30. Sugar Tooth Candy Company needs 300 gallons of a 32% sucrose solu-
tion for a certain kind of candy. The company has a solution that is 60% A la:. sucrose and a solution that is 25% sucrose. How many gallons of each .;i.icrose; 240 should the company mix together to obtain the desired solution? 3f 25% sucrosa
31. Mr. Arnold is going to invest $1550 in two separate accounts. One account pays 7.5%, and the other account pays 8.25%. How much should he invest in each account so that his annual return on his investment will be $121.20? .1390 at 7.3%; 3:360 et 3,2511
32. Two fishing boats leave Sandy Cove at the same time traveling in the same direction. One boat is traveling three times as fast as the other boat. Af:er five hours the faster boat is 80 miles ahead of the slower boat. What is the speed of each boat? slow an y. last one, 24 mph
Graph each system of inequalities. 33. 2x + y < 6 35. 3x + 5y 12
3x - y 4 2x - 3y < -6 34. 5x + 2y 8 36. y 3x + 4
x y > -5 y > 2
37. What does the graph of an inconsistent linear system look like? 2 parallel Ines 38. Explain the mathematical significance of II Corinthians 5:21. 2hrist died
as t, substitute in our *ca. His payment for Our debt is an illustration of the priticiple of substitution.