chapter 7 systems of linear equations. a) 2x + 3y = 3 b) 4x – 2y = 14 x + y = 4 2x + y = 7 c) 2x...

Chapter 7

Systems of Linear Equations

a) 2x + 3y = 3 b) 4x – 2y = 14 x + y = 4 2x + y = 7

c) 2x – 3y = 3 d) 2x – 3y = 9 -x + 6y = -3 -x + 6y = -9

1. Which linear system has the solution x = 3 and y = -1?

3 + (-1) = 4 4(3) – 2(-1) = 14

12 + 2 = 14

2(3) + 3(-1) = 3 6 – 3 = 3

2(3) – 3(-1) = 3

6 – (-3) = 3

2(3) – 3(-1) = 9

9 = 9

6 + 3 = 9

-3 + 6(-1) = -9

-9 = -9

-3 + (-6) = -9

3 = 3 14 = 14

2(3) + (-1) = 7 6 – 1 = 7

2. a) Create a linear system to model this situation.

2 jackets and 2 sweaters cost $228. A jacket costs $44 more than a sweater

Cost jackets = x Cost sweaters = y

2x + 2y = 228x – y = 44

b) Kurt has determined that a sweater costs $35 and jackets cost $79. Use the linear system from part a to verify that he is correct.

2(79) + 2(35) = 228158 + 70 = 228

228 = 228

79 – 35 = 4444 = 44

3. Solve this linear system by graphing.2x + y = 73x + 3y = 6

y = mx + b12

1 2x + y = 7-2x -2x

y = -2x +7

2 3x + 3y = 6-3x -3x

3y = -3x + 63 3 3 y = -x + 2

(5, -3) x = 5 y = -3

4. Determine the number of solutions to the linear system-6x + 2y = -4 3x – y = 2

DO NOT SOLVE FOR X AND Y

Compare slope and y-intercept:

1 – If m and b are the same Infinite Solutions2 – If only m is the same No Solutions3 – If m and b are different One Solution

y = mx + b

12

1 -6x + 2y = -4+6x +6x

2y = 6x – 42 2 2

y = 3x – 2

m = 3 b = -2

2 3x – y = 2-3x -3x

-y = -3x + 2-1 -1 -1

y = 3x – 2 m = 3 b = -2

Infinite Solutions

5. Solve the following system using both Substitution and Elimination

244

33 yx

1223

2 yx

Get rid of fractions!

1

2

244

33 yx1 4( )

96312 yx

1223

2 yx2 3( )

3662 yx

5. (part 2) Solve the following system using both Substitution and elimination 1

2

3662 yx2-6y -6y

96312 yx

3662 yx2 2 2

3662 yxSubstitution:

183 yx

96312 yx1963)183(12 yy

y36 216 y3 96

y39 96216

31239 y3939 8y

18)8(3 x1824 x6x

216216

5. (part 3) Solve the following system using both Substitution and elimination 1

2

3662 yx6( )

96312 yx

x12

3662 yxElimination:

0 y39 312

8y

96312 yx

y36 216

-39 -39

2 3662 yx

36482 x4848

122 x 2 2

6x

36)8(62 x

6. Sam scored 80% on part of A of a math test and 92% on part B of the math test. His total mark for the test was 63. The total mark possible for the test was 75. How many marks is each part worth?

Solve the following system using both Substitution and elimination

75ba

6392.08.0 ba

Create system of linear equations

1

2

let a = number of marks on part A b = number of marks on part B

6. (part 2) Solve the following system using both Substitution and elimination 1

2

75ba1-b -b

ba 75

Substitution:

6392.08.0 ba26392.0)75(8.0 bb

60 b8.0 b92.0 63

60 6312.0 b

312.0 b12.012.0

25b

2575 a

50a

6060

75ba6392.08.0 ba

Part A is out of 50, Part B is out of 25

6. (part 3) Solve the following system using both Substitution and elimination 1

275ba0.8( )

a8.0

Elimination:

0 b12.0 3

25b

6392.08.0 ba

b8.0 60

0.12 0.12

1 75ba

7525a2525 50a

75ba6392.08.0 ba

Part A is out of 50, Part B is out of 25