TOPIC VOCABULARY

• consistentsystem, p. 70

• constraint, p. 88

• coordinatespace, p. 94

•dependentsystem, p. 70

•equivalentsystems, p. 77

• feasibleregion, p. 88

•Gaussianelimination, p. 102

• inconsistentsystem, p. 70

• independentsystem, p. 70

• linearprogramming, p. 88

• linearsystem, p. 70

•matrix, p. 102

•matrixelement, p. 102

•objectivefunction, p. 88

•orderedtriple, p. 94

• rowoperation, p. 102

• solutionofasystem, p. 70

• systemofequations, p. 70

Check Your UnderstandingChoose the correct term to complete each sentence.

1. A consistent system with exactly one solution is a(n) ? .

2. ? is a method for finding a minimum or maximum value, given a system of limits called ? .

Topic 3 Review

3-1 Solving Systems Using Tables and Graphs

ExercisesWithout graphing, classify each system of equations as independent, dependent, or inconsistent. Solve independent systems by graphing.

3. e6x - 2y = 2

2 + 6x = y 4. e5 - y = 2x

6x - 15 = -3y

5. e 6y + 2x = 8

12y + 4x = 4 6. e1.5 + 3x = 0.5y

6 - 2y = -12x

7. e2 - 0.25x = 0.5y-1.5y = 1.5x - 3

8. e1 + y = xx + y = 1

9. For $7.52, you purchased 8 pens and highlighters from a local bookstore. Each highlighter cost $1.09 and each pen cost $.69. How many pens did you buy?

ExampleSolve the system e3x + 2y = 4

2x - 4y = 8 and graph the equations.

The only solution, where the lines intersect, is (2, -1).

�2

�4

2

O�4

y

x

Quick ReviewA system of equations has two or more equations. Points of intersection are solutions. A linear system has linear equations. A consistent system can be dependent, with infinitely many solutions, or independent, with one solution. An inconsistent system has no solution.

108 Topic 3 Review

3-3 Systems of Inequalities

ExercisesSolve each system of inequalities by graphing.

15. ey 6 4x3x + y Ú 5

16. •2x + 3y 7 6

x … -1

y Ú 4

17. For a community breakfast there should be at least three times as much regular coffee as decaffeinated coffee. A total of ten gallons is sufficient for the breakfast. Write and graph a system of inequalities to model the problem.

ExampleSolve the system of inequalities by graphing.

ey 7 -3

y … -x - 1

Graph both inequalities and shade the region valid for both inequalities.

xy

O42−2

−4

Quick ReviewTo solve a system of inequalities by graphing, first graph the boundaries for each inequality. Then shade the region(s) of the plane containing solutions valid for both inequalities.

3-2 Solving Systems Algebraically

ExercisesSolve each system by substitution.

10. e x - 2y = 3

3x + y = -5 11. e14x - 35 = 7y

-25 - 6x = 5y

Solve each system by elimination.

12. e11 - 5y = 2x5y + 3 = -9x

13. e2x + 3y = 4

4x + 6y = 9

14. Roast beef has 25 g of protein and 11 g of calcium per serving. A serving of mashed potatoes has 2 g of protein and 25 g of calcium. How many servings of each are needed to supply exactly 29 g of protein and 61 g of calcium?

Example

Solve e10 - y = 4xx = 4 + 0.5y

by substitution.

Substitute for x: 10 - y = 4(4 + 0.5y) = 16 + 2y.

Solve for y: y = -2.

Substitute into the first equation: 10 - (-2) = 4x.

Solve for x: x = 3. The solution is (3, -2).

Quick ReviewTo solve an independent system by substitution, solve one equation for a variable. Then substitute that expression into the other equation and solve for the remaining variable. To solve by elimination, add two equations with additive inverses as coefficients to eliminate one variable and solve for the other. In both cases you solve for one of the variables and use substitution to solve for the remaining variable.

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3-4 Linear Programming

ExercisesGraph the system of constraints. Name the vertices. Then find the values of x and y that maximize or minimize the objective function.

18. •x Ú 2

y Ú 0

3x + 2y Ú 12

19. •3x + 2y … 12

x + y … 5

x Ú 0, y Ú 0

Minimum for Maximum for C = x + 5y P = 3x + 5y

20. A lunch stand makes $.75 profit on each chef’s salad and $1.20 profit on each Caesar salad. On a typical weekday, it sells between 40 and 60 chef’s salads and between 35 and 50 Caesar salads. The total number sold has never exceeded 100 salads. How many of each type should be prepared in order to maximize profit?

ExampleGraph the system of constraints and name the vertices. •

x … 8

y … 5

x Ú 0, y Ú 0Objective function: P = 2x + y

Graph the inequalities and shade the area satisfying all inequalities.

The vertices of the feasible region are (0, 0), (0, 5), (8, 5), and (8, 0).

Evaluate the objective function at each vertex:

2(0) + 0 = 0

2(8) + 5 = 21

2(0) + 5 = 5

2(8) + 0 = 16

The maximum value occurs at (8, 5).

2

2

4

6

4 6O

y

x

(0, 5) (8, 5)

(8, 0)(0, 0)

Quick ReviewLinear programming is used to find a minimum or maximum of an objective function, given constraints as linear inequalities. The maximum or minimum occurs at a vertex of the feasible region, which contains the solutions to the system of constraints.

110 Topic 3 Review

3-5 Systems in Three Variables

ExercisesSolve each system by elimination.

21. c x + y - 2z = 8

5x - 3y + z = -6

-2x - y + 4z = -13

22. c -x + y + 2z = -5

5x + 4y - 4z = 4

x - 3y - 2z = 3

Solve each system by substitution.

23. •3x + y - 2z = 22

x + 5y + z = 4

x = -3z

24. •x + 2y + z = 14

y = z + 1

x = -3z + 6

ExampleSolve by elimination.

①

②

③

c x + y + z = 10

2x - y + z = 9

-3x + 2y + 2z = 5

Add equations ① and ② to eliminate y. ④ 3x + 2z = 19

Add 2 times ② to ③ to eliminate y. ⑤ x + 4z = 23

Add -3 times ⑤ to ④ to eliminate x. z = 5

Substitute z = 5 into ⑤. x = 3

Substitute z = 5 and x = 3 into ① or ②. y = 2

The solution to the system is (3, 2, 5).

Quick ReviewTo solve a system of three equations, either pair two equations and eliminate the same variable from both equations, using one equation twice, or choose an equation, solve for one variable, and substitute the expression for that variable into the other two equations. Then, solve the remaining system.

3-6 Solving Systems Using Matrices

ExercisesSolve each system using a matrix.

25. b4x - 12y = -1

6x + 4y = 4

26. b 7x + 2y = 5

13x + 14y = -1

27. c -5x + 3y + 4z = 2

3x - y - z = 4

x - 6y - 5z = -4

28. • x + y + z = 4

2x - y + z = 5

x + y - 2z = 13

ExampleSolve using a matrix. e6x + 3y = -15

2x + 4y = 10

Enter coefficients as matrix elements J6 3

2 4 ` -15

10R .

Divide the first row by 3 to get J2 1

2 4 ` -5

10R . Subtract the

first row from the second row to get J2 1

0 3 ` -5

15R . Multiply

the second row by 13 to get J2 1

0 1 ` -5

5R . Subtract the second

row from the first row to get J2 0

0 1 ` -10

5R . Divide the first row

by 2 to get J1 0

0 1 ` -5

5R . The solution to the system is (-5, 5).

Quick ReviewA matrix can represent a system of equations where each row stands for a different equation. The columns contain the coefficients of the variables and the constants.

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