Question

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Explanation:

Step1: Solve Problem 1

We have the system:
\( y = 6 \)
\( -x + y = 2 \)

Substitute \( y = 6 \) into the second equation:
\( -x + 6 = 2 \)
Subtract 6 from both sides:
\( -x = 2 - 6 \)
\( -x = -4 \)
Multiply both sides by -1:
\( x = 4 \)

So the solution is \( (4, 6) \).

Step2: Solve Problem 2

We have the system:
\( x = 5 \)
\( 3x - 2y = 9 \)

Substitute \( x = 5 \) into the second equation:
\( 3(5) - 2y = 9 \)
\( 15 - 2y = 9 \)
Subtract 15 from both sides:
\( -2y = 9 - 15 \)
\( -2y = -6 \)
Divide both sides by -2:
\( y = 3 \)

So the solution is \( (5, 3) \).

Step3: Solve Problem 3

We have the system:
\( 4x = -8 \)
\( 2x + 5y = 6 \)

First, solve the first equation for \( x \):
\( x = \frac{-8}{4} = -2 \)

Substitute \( x = -2 \) into the second equation:
\( 2(-2) + 5y = 6 \)
\( -4 + 5y = 6 \)
Add 4 to both sides:
\( 5y = 6 + 4 \)
\( 5y = 10 \)
Divide both sides by 5:
\( y = 2 \)

So the solution is \( (-2, 2) \).

Step4: Solve Problem 4

We have the system:
\( y = -5x \)
\( 6x + y = 2 \)

Substitute \( y = -5x \) into the second equation:
\( 6x + (-5x) = 2 \)
\( 6x - 5x = 2 \)
\( x = 2 \)

Now substitute \( x = 2 \) into \( y = -5x \):
\( y = -5(2) = -10 \)

So the solution is \( (2, -10) \).

Step5: Solve Problem 5

We have the system:
\( x = 9y \)
\( 3y - 3x = 24 \)

Substitute \( x = 9y \) into the second equation:
\( 3y - 3(9y) = 24 \)
\( 3y - 27y = 24 \)
\( -24y = 24 \)
Divide both sides by -24:
\( y = -1 \)

Now substitute \( y = -1 \) into \( x = 9y \):
\( x = 9(-1) = -9 \)

So the solution is \( (-9, -1) \).

Step6: Solve Problem 6

We have the system:
\( y - x = -8 \)
\( 2x - 16 = y \)

From the second equation, \( y = 2x - 16 \). Substitute into the first equation:
\( (2x - 16) - x = -8 \)
\( 2x - 16 - x = -8 \)
\( x - 16 = -8 \)
Add 16 to both sides:
\( x = -8 + 16 \)
\( x = 8 \)

Now substitute \( x = 8 \) into \( y = 2x - 16 \):
\( y = 2(8) - 16 \)
\( y = 16 - 16 \)
\( y = 0 \)

So the solution is \( (8, 0) \).

Step7: Solve Problem 7

We have the system:
\( y = 3x + 1 \)
\( y = 8x - 4 \)

Set the two equations equal to each other:
\( 3x + 1 = 8x - 4 \)
Subtract \( 3x \) from both sides:
\( 1 = 5x - 4 \)
Add 4 to both sides:
\( 5 = 5x \)
Divide both sides by 5:
\( x = 1 \)

Now substitute \( x = 1 \) into \( y = 3x + 1 \):
\( y = 3(1) + 1 \)
\( y = 3 + 1 \)
\( y = 4 \)

So the solution is \( (1, 4) \).

Step8: Solve Problem 8

We have the system:
\( x + y = 0 \)
\( 5x + 4y = 6 \)

From the first equation, \( y = -x \). Substitute into the second equation:
\( 5x + 4(-x) = 6 \)
\( 5x - 4x = 6 \)
\( x = 6 \)

Now substitute \( x = 6 \) into \( y = -x \):
\( y = -6 \)

So the solution is \( (6, -6) \).

Step9: Solve Problem 9

We have the system:
\( 2y = -2x + 6 \)
\( x - 5y = -15 \)

Simplify the first equation by dividing by 2:
\( y = -x + 3 \)

Substitute \( y = -x + 3 \) into the second equation:
\( x - 5(-x + 3) = -15 \)
\( x + 5x - 15 = -15 \)
\( 6x - 15 = -15 \)
Add 15 to both sides:
\( 6x = 0 \)
Divide both sides by 6:
\( x = 0 \)

Now substitute \( x = 0 \) into \( y = -x + 3 \):
\( y = -0 + 3 \)
\( y = 3 \)

So the solution is \( (0, 3) \).

Step10: Solve Problem 10

We have the system:
\( 5x + y = 2 \)
\( 20x + 3y = -4 \)

From the first equation, \( y = 2 - 5x \). Substitute into the second equation:
\( 20x + 3(2 - 5x) = -4 \)
\( 20x + 6 - 15x = -4 \)
\( 5x + 6 = -4 \)
Subtract 6 from both sides:
\( 5x = -10 \)
Divide both sides by 5:
\( x = -2 \)

Now substitute \( x = -2 \) into \( y = 2 - 5x \):
\( y = 2 - 5(-2) \)
\( y = 2 + 10 \)
\( y = 12 \)

So the solution is \( (-2, 12) \).

Step11: Solve Problem 11

We have the system:
\( -8x + 2y = -14 \)
\( 4x - 6y = 22 \)

Simplify the first equation by dividing by 2:
\( -4x + y = -7 \)
So \( y = 4x - 7 \)

Substitute \( y = 4x - 7 \) into the second equation:
\( 4x - 6(4x - 7) = 22 \)
\( 4x - 24x + 42 = 22 \)
\( -20x + 42 = 22 \)
Subtract 42 from both sides:
\( -20x = 22 - 42 \)
\( -20x = -20 \)
Divide both sides by -20:
\( x = 1 \)

Now substitute \( x = 1 \) into \( y = 4x - 7 \):
\( y = 4(1) - 7 \)
\( y = 4 - 7 \)
\( y = -3 \)

So the solution is \( (1, -3) \).

Step12: Solve Problem 12

We have the system:
\( 2x + 4y = -18 \)
\( 5x - 10y = -5 \)

Simplify the first equation by dividing by 2:
\( x + 2y = -9 \)
So \( x = -9 - 2y \)

Substitute \( x = -9 - 2y \) into the second equation:
\( 5(-9 - 2y) - 10y = -5 \)
\( -45 - 10y - 10y = -5 \)
\( -45 - 20y = -5 \)
Add 45 to both sides:
\( -20y = -5 + 45 \)
\( -20y = 40 \)
Divide both sides by -20:
\( y = -2 \)

Now substitute \( y = -2 \) into \( x = -9 - 2y \):
\( x = -9 - 2(-2) \)
\( x = -9 + 4 \)
\( x = -5 \)

So the solution is \( (-5, -2) \).

Answer:

1. \( (4, 6) \)
2. \( (5, 3) \)
3. \( (-2, 2) \)
4. \( (2, -10) \)
5. \( (-9, -1) \)
6. \( (8, 0) \)
7. \( (1, 4) \)
8. \( (6, -6) \)
9. \( (0, 3) \)
10. \( (-2, 12) \)
11. \( (1, -3) \)
12. \( (-5, -2) \)