Algebra, Trig Solve the System: 7x +

Algebra, Trig

Solve the system: 7x + 3y = -2, -7x -- 7y =

(7x + 3y) + (-7x -- 7y) = (-2 + 14) 7x + 3y -- 7x -- 7y = 12 -4y = 12 y = -3

Substituting y for the first equation: 7x + 3(-3) = -2 7x -- 9 = -2 7x = 7 x = 1

x = 1, y = -3.

Solve the system: x + y = -5, x -- y = 12

(x + y) + (x -- y) = (-5 + 12) x + y + x -- y = 7 2x = 7 x = 7/2

Substituting x for the first equation: 7/2 + y = -5 y = -5 -- (7/2) y = -17/2

x = 7/2, y = -17/2.

Solve the system: y -- 3z = -12, -2x + y + 2z = 5, 2x + 3z = 7

(y -- 3z) + (-2x + y + 2z) + (2x + 3z) = -12 + 5 + 7 y -- 3z -- 2x + y + 2z + 2x + 3z = 0

2z = 0 y + z = 0 y = -z

Substituting for y for the first equation: -z -- 3z = -12 -4z = -12 z = 3 y = -3

Substituting z for the third equation: 2x + 3(3) = 7 2x + 9 = 7 2x = -2 x = -1

Answer: x = -1, y = -3, z = 3.

Solve the system: y = 4x + 3, 4y -- 20x = -8

4y -- 20x = -8 4y = 20x -- 8 y = 5x --

4x + 3 = 5x -- 2 x = 5

Substituting x for the first equation: y = 4(5) + 3 = 20 + 3 = 23

Answer: x = 5, y = 23.

5. Solve the system: x -- y + z = 6, x + y + z = 2, x + y -- z =0

x + y -- z = 0 x + y = z

Substituting z for the second equation: x + y + 2 = 2 z + z = 2 2z = 2 z = 1

(x -- y + z) + (x + y -- z) = (6 + 0) 2x = 6 x = 3

Substituting x and z for the third equation: 3 + y -- 1 = 0 y + 2 = 0 y = -2

Answer: x = 3, y = -2, z = 1.

6. Solve the system: x -- y + 2z = 7, 2x + z = 4, x + 5y + z = 9

2x + z = 4 z = 4 -- 2x

Substitute z for first equation: x -- y + 2(4 -- 2x) = 7 x -- y + 8 -- 4x = 7 -3x -- y = -1

Substitute z for third equation: x + 5y + (4 -- 2x) = 9 x + 5y + 4 -- 2x = 9 -x + 5y = 5

-x + 5y = 5 x = 5y -- 5

Substitute x in -3x -- y = -1 equation: -3(5y -- 5) -- y = -1 -15y + 15 -- y = -1

-16y = -16 y = 1

Substituting y for -3x -- y = 1 -3x -- 1 = -1 -3x = 0 x = 0

Substituting x for second equation: 2(0) + z = 4 z = 4

Answer: x = 0, y = -1, z = 4.

7. Solve the system: 3x + 4y = 40, x -- 2y = 0

2(x -- 2y) = 0 2x -- 4y = 0

(3x + 4y) + (2x -- 4y) = 40 + 0 5x = 40 x = 8

Substituting x for the first equation: 3(8) + 4y = 40-24 + 4y = 40 4y = 16 y = 4

Answer: x = 8, y = 4.

8. Solve the system: x + 6y = 24, -6x + 5y = -21

6(x + 6y = 24) 6x + 36y = 144

(6x + 36y) + (-6x + 5y) = 144 + (-21) 6x + 36y -- 6x + 5y = 123 41y = 123 y = 3

Substituting y for the first equation: x + 6(3) = 24 x + 18 = 24 x = 6

Answer: x = 6, y = 3.

9. Solve the system: x + y + z= -1, x -- y + 3z = -17, 2x + y + z = -2

Equation 1: (x + y + z) + (x -- y + 3z) = -1 + (-17) 2x + 4z = -18

Equation 2: (x -- y + 3z) + (2x + y + z) = -17 + (-2) 3x + 4z = -19

Equation 2 -- Equation 1: (3x + 4z) -- (2x + 4z) = -19 -- (-18) x = -1

Substituting x into Equation 1: 2(-1) + 4z = -18 -2 + 4z = -18 4z = -16 z = -4

Substituting x and z into first equation: -1 + y + (-4) = -1 -5 + y = -1 y = 4

Answer: x = -1, y = 4, z = -4.

10. Solve the system: 3x + 4y = -2, x = -2y

Substitute x for the first equation: 3(-2y) + 4y = -2 -6y + 4y = -2 -2y = -2 y = 1

Substitute y for the second equation: x = -2(1) = -2

Answer: x = -2, y = 1.

11. The ordered pair: (5,6) is a solution of the following system:

x + y = 11, x -- y = -1 -- True or false

Checking:

(x, y) = (5, 6) x + y = 11 5 + 6 = 11 = 11

(x, y) = (5, 6) x -- y = -1-5 -- 6 = -1 -1 = -1

Answer: TRUE.

12. Solve the system: x + y = 10, y = 4x

Substituting x for the first equation: x + 4x = 10 5x = 10 x = 2

Substituting x for the second equation: y = 4(2) y = 8

Answer: x = 2, y = 8.

13. Solve the system: -32 + 3z = 4(x -- 2y), 4(x -- 2y -- z) = - 36, -2(2x + y) + 2z = -12

Simplifying equations:

-32 + 3z = 4(x -- 2y) -32 + 3z = 4x -- 8y 4x -- 8y -- 3z = -32

4(x -- 2y -- z) = -36 4x -- 8y -- 4z = -36 x -- 2y -- z = -9

-2(2x + y) + 2z = -12 -4x -- 2y + 2z = -12 2x + y -- z = 3