Solve each system of linear equations by graphing on the provided coordinate grids.

To solve each system of equations by graphing, we need to find the point where the two lines intersect. This point (x, y) is the solution because it satisfies both equations at the same time.

Let’s go through each problem one by one:

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Problem 1:
Equations:
- 2x + 3y = -6
- -2x + 3y = 6

Step 1: Find intercepts for first equation (2x + 3y = -6)
- When x=0 → 3y=-6 → y=-2 → (0, -2)
- When y=0 → 2x=-6 → x=-3 → (-3, 0)

Step 2: Find intercepts for second equation (-2x + 3y = 6)
- When x=0 → 3y=6 → y=2 → (0, 2)
- When y=0 → -2x=6 → x=-3 → (-3, 0)

Wait — both lines pass through (-3, 0)? Let’s check if they’re different lines.

Actually, let’s add the two equations:
(2x + 3y) + (-2x + 3y) = -6 + 6 → 6y = 0 → y = 0

Plug y=0 into first equation: 2x + 0 = -6 → x = -3

So intersection is at (-3, 0). That’s the solution.

✔ Solution: (-3, 0)

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Problem 2:
Equations:
- -2x + 3y = -9
- x + y = 2

Step 1: Solve second equation for y: y = 2 - x

Step 2: Plug into first equation:
-2x + 3(2 - x) = -9
→ -2x + 6 - 3x = -9
→ -5x = -15 → x = 3

Then y = 2 - 3 = -1

Check in first equation: -2(3) + 3(-1) = -6 -3 = -9 ✔

✔ Solution: (3, -1)

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Problem 3:
Equations:
- -x + 3y = 6
- 4x + 3y = -9

Subtract first from second:
(4x + 3y) - (-x + 3y) = -9 - 6
→ 4x + 3y + x - 3y = -15
→ 5x = -15 → x = -3

Plug into first equation: -(-3) + 3y = 6 → 3 + 3y = 6 → 3y = 3 → y = 1

Check in second: 4(-3) + 3(1) = -12 + 3 = -9 ✔

✔ Solution: (-3, 1)

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Problem 4:
Equations:
- -x + 3y = -6 ← Note: original says “-x +3y -6” which is likely typo; should be “= -6”
- -5x + 3y = 6

Subtract first from second:
(-5x + 3y) - (-x + 3y) = 6 - (-6)
→ -5x + 3y + x - 3y = 12
→ -4x = 12 → x = -3

Plug into first: -(-3) + 3y = -6 → 3 + 3y = -6 → 3y = -9 → y = -3

Check in second: -5(-3) + 3(-3) = 15 - 9 = 6 ✔

✔ Solution: (-3, -3)

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Problem 5:
Equations:
- -5x + 4y = -16
- x + 4y = 8

Subtract second from first:
(-5x + 4y) - (x + 4y) = -16 - 8
→ -6x = -24 → x = 4

Plug into second: 4 + 4y = 8 → 4y = 4 → y = 1

Check in first: -5(4) + 4(1) = -20 + 4 = -16 ✔

✔ Solution: (4, 1)

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Problem 6:
Equations:
- 2x + y = 2
- -x + y = -4

Subtract second from first:
(2x + y) - (-x + y) = 2 - (-4)
→ 2x + y + x - y = 6
→ 3x = 6 → x = 2

Plug into second: -2 + y = -4 → y = -2

Check in first: 2(2) + (-2) = 4 - 2 = 2 ✔

✔ Solution: (2, -2)

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Problem 7:
Equations:
- 5x + 3y = 12
- -x + 3y = -6

Subtract second from first:
(5x + 3y) - (-x + 3y) = 12 - (-6)
→ 5x + 3y + x - 3y = 18
→ 6x = 18 → x = 3

Plug into second: -3 + 3y = -6 → 3y = -3 → y = -1

Check in first: 5(3) + 3(-1) = 15 - 3 = 12 ✔

✔ Solution: (3, -1)

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Problem 8:
Equations:
- -x + 2y = -6
- 3x + 2y = 2

Subtract first from second:
(3x + 2y) - (-x + 2y) = 2 - (-6)
→ 3x + 2y + x - 2y = 8
→ 4x = 8 → x = 2

Plug into first: -2 + 2y = -6 → 2y = -4 → y = -2

Check in second: 3(2) + 2(-2) = 6 - 4 = 2 ✔

✔ Solution: (2, -2)

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Final Answers:

1) (-3, 0)
2) (3, -1)
3) (-3, 1)
4) (-3, -3)
5) (4, 1)
6) (2, -2)
7) (3, -1)
8) (2, -2)