SOLUTION: Solving systems of equations by graphing - Studypool - Free Printable

You're asked to solve each system of equations by graphing. That means:

1. Graph both lines on the coordinate plane.
2. Find the point where they intersect — that’s the solution (x, y).
3. If the lines are parallel → no solution.
4. If the lines are the same → infinitely many solutions.

Since I can’t draw graphs here, I’ll solve each system algebraically (using substitution or elimination) to find the exact intersection point — which is what you’d see if you graphed them correctly. Then I’ll describe how to plot them for verification.

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✔ 1)

Equations:
`2x + 3y = -6`
`-2x + 3y = 6`

Add the equations to eliminate x:
(2x + 3y) + (-2x + 3y) = -6 + 6
→ `6y = 0` → `y = 0`

Plug into first equation:
2x + 3(0) = -6 → 2x = -6 → `x = -3`

✔ Solution: (-3, 0)

*Graphing tip:*
- Line 1: 2x + 3y = -6 → intercepts: x=-3, y=-2
- Line 2: -2x + 3y = 6 → intercepts: x=-3, y=2
They cross at (-3, 0)

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✔ 2)

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

Solve second for x: `x = 2 - y`

Substitute into first:
-2(2 - y) + 3y = -9
→ -4 + 2y + 3y = -9
→ 5y = -5 → `y = -1`

Then x = 2 - (-1) = 3

✔ Solution: (3, -1)

*Graphing tip:*
- Line 1: -2x + 3y = -9 → x-intercept: 4.5, y-intercept: -3
- Line 2: x + y = 2 → intercepts: (2,0), (0,2)
Cross at (3, -1)

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✔ 3)

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

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

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

✔ Solution: (-3, 1)

*Graphing tip:*
- Line 1: -x + 3y = 6 → y = (x+6)/3 → points: (0,2), (-3,1)
- Line 2: 4x + 3y = -9 → y = (-4x -9)/3 → points: (0,-3), (-3,1)
Cross at (-3,1)

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❗ 4)

Equations:
`-x + 3y = -6` ← (I assume you meant “= -6”, since it says “-x +3y -6” — probably a typo)
`-5x + 3y = 6`

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

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

✔ Solution: (-3, -3)

*Graphing tip:*
- Line 1: -x + 3y = -6 → y = (x -6)/3 → points: (0,-2), (-3,-3)
- Line 2: -5x + 3y = 6 → y = (5x +6)/3 → points: (0,2), (-3,-3)
Cross at (-3,-3)

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✔ 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`

✔ Solution: (4, 1)

*Graphing tip:*
- Line 1: -5x + 4y = -16 → y = (5x -16)/4 → points: (0,-4), (4,1)
- Line 2: x + 4y = 8 → y = (8 - x)/4 → points: (0,2), (4,1)
Cross at (4,1)

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✔ 6)

Equations:
`2x + y = 2`
`-x + y = -4`

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

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

✔ Solution: (2, -2)

*Graphing tip:*
- Line 1: 2x + y = 2 → y = -2x + 2 → points: (0,2), (1,0), (2,-2)
- Line 2: -x + y = -4 → y = x -4 → points: (0,-4), (4,0), (2,-2)
Cross at (2,-2)

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✔ 7)

Equations:
`5x + 3y = 12`
`-x + 3y = -6`

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

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

✔ Solution: (3, -1)

*Graphing tip:*
- Line 1: 5x + 3y = 12 → y = (12 -5x)/3 → points: (0,4), (3,-1)
- Line 2: -x + 3y = -6 → y = (x -6)/3 → points: (0,-2), (3,-1)
Cross at (3,-1)

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✔ 8)

Equations:
`-x + 2y = -6`
`3x + 2y = 2`

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

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

✔ Solution: (2, -2)

*Graphing tip:*
- Line 1: -x + 2y = -6 → y = (x -6)/2 → points: (0,-3), (2,-2)
- Line 2: 3x + 2y = 2 → y = (2 -3x)/2 → points: (0,1), (2,-2)
Cross at (2,-2)

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## 📌 Final Answers:

1) (-3, 0)
2) (3, -1)
3) (-3, 1)
4) (-3, -3) *(assuming typo corrected)*
5) (4, 1)
6) (2, -2)
7) (3, -1)
8) (2, -2)

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📌 To graph each system:

For each pair:
- Convert to slope-intercept form (`y = mx + b`) if needed.
- Plot two points per line (often x and y intercepts).
- Draw the lines.
- Mark the intersection point — that’s your solution!

Let me know if you want step-by-step graphing instructions for any one problem!