Graphing Systems of Equations Worksheets - Math Monks - Free Printable

Let’s solve each system of equations by finding where the two lines intersect — that point (x, y) is the solution.

We’ll do this algebraically first to find the exact solution, then you can graph it if needed. But since the problem says “graph each system and identify its solution,” we’ll find the intersection point mathematically so you know what to look for on the graph.

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Problem ①:
y = -¼x + 3
y = -x + 6

Set them equal:
-¼x + 3 = -x + 6

Add x to both sides:
(-¼x + x) + 3 = 6 → (¾x) + 3 = 6

Subtract 3:
¾x = 3

Multiply both sides by 4/3:
x = 3 × (4/3) = 4

Plug into either equation, say y = -x + 6:
y = -4 + 6 = 2

✔ Solution: (4, 2)

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Problem ②:
y = -½x + 2
y = -⁴₂x - 1 → simplify: -⁴⁄₂ = -2 → y = -2x - 1

Set equal:
-½x + 2 = -2x - 1

Add 2x to both sides:
(-½x + 2x) + 2 = -1 → (³⁄₂x) + 2 = -1

Subtract 2:
³⁄₂x = -3

Multiply both sides by 2/3:
x = -3 × (2/3) = -2

Plug into y = -2x - 1:
y = -2(-2) - 1 = 4 - 1 = 3

✔ Solution: (-2, 3)

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Problem ③:
y = x - 4
y = -x + 2

Set equal:
x - 4 = -x + 2

Add x to both sides:
2x - 4 = 2

Add 4:
2x = 6 → x = 3

Plug into y = x - 4:
y = 3 - 4 = -1

✔ Solution: (3, -1)

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Problem ④:
y = ⅓x - 3
y = -x + 1

Set equal:
⅓x - 3 = -x + 1

Add x to both sides:
⅓x + x - 3 = 1 → (⁴⁄₃x) - 3 = 1

Add 3:
⁴⁄₃x = 4

Multiply both sides by 3/4:
x = 4 × (3/4) = 3

Plug into y = -x + 1:
y = -3 + 1 = -2

✔ Solution: (3, -2)

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Problem ⑤:
y = 2 - 2x → same as y = -2x + 2
y = -4x + 1

Set equal:
-2x + 2 = -4x + 1

Add 4x to both sides:
2x + 2 = 1

Subtract 2:
2x = -1 → x = -½

Plug into y = -2x + 2:
y = -2(-½) + 2 = 1 + 2 = 3

✔ Solution: (-½, 3) or (-0.5, 3)

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Problem ⑥:
First equation: y - 3x = 4 → rewrite as y = 3x + 4
Second equation: y = 3x + 2

Wait — both have slope 3? Let’s check:

y = 3x + 4
y = 3x + 2

These are parallel lines! Same slope, different y-intercepts → they never intersect.

So no solution.

But let’s double-check by setting equal:

3x + 4 = 3x + 2
Subtract 3x from both sides:
4 = 2 → FALSE

No solution exists.

✔ Solution: No solution (parallel lines)

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Final Answer:

① (4, 2)
② (-2, 3)
③ (3, -1)
④ (3, -2)
⑤ (-0.5, 3)
⑥ No solution