Solving Systems of Equations by Substitution Worksheets - Math Monks - Free Printable

Let’s solve each system of equations by graphing. Since we’re doing this without actual graph paper, I’ll find the solution algebraically (by substitution or elimination) — which gives us the same point where the two lines would intersect on a graph.

We’ll go one by one.

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1)
2x + y = 3
x - 3y = 4

Solve first equation for y:
y = 3 - 2x

Plug into second:
x - 3(3 - 2x) = 4
x - 9 + 6x = 4
7x = 13 → x = 13/7 ≈ 1.857
Then y = 3 - 2*(13/7) = 3 - 26/7 = (21 - 26)/7 = -5/7 ≈ -0.714

✔ Solution: (13/7, -5/7)

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2)
y = -2
4x - 3y = 18

Plug y = -2 into second:
4x - 3(-2) = 18 → 4x + 6 = 18 → 4x = 12 → x = 3

✔ Solution: (3, -2)

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3)
3x + y = 3
6x + 2y = 4

Notice: Second equation is almost double the first? Let’s check:
Double first: 6x + 2y = 6 → but second says 6x + 2y = 4 → contradiction!
So no solution — parallel lines.

✔ Solution: No solution

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4)
4x + y = 2
x - y = 3

Add both equations:
(4x + y) + (x - y) = 2 + 3 → 5x = 5 → x = 1
Then from x - y = 3 → 1 - y = 3 → y = -2

✔ Solution: (1, -2)

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5)
y = (1/2)x - 1
y = (1/4)x - 4

Set equal:
(1/2)x - 1 = (1/4)x - 4
Multiply all terms by 4 to eliminate fractions:
2x - 4 = x - 16
2x - x = -16 + 4 → x = -12
Then y = (1/2)(-12) - 1 = -6 -1 = -7

✔ Solution: (-12, -7)

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6)
y = 4x + 3
y = x - 2

Set equal:
4x + 3 = x - 2 → 3x = -5 → x = -5/3
Then y = (-5/3) - 2 = -5/3 - 6/3 = -11/3

✔ Solution: (-5/3, -11/3)

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7)
y = -2x + 2
y = -2x - 2

Same slope, different y-intercepts → parallel lines → no solution

✔ Solution: No solution

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8)
x - y = 3
2x + y = 0

Add them:
(x - y) + (2x + y) = 3 + 0 → 3x = 3 → x = 1
Then from x - y = 3 → 1 - y = 3 → y = -2

✔ Solution: (1, -2)

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9)
y = x + 1
2y = 3x → y = (3/2)x

Set equal:
x + 1 = (3/2)x
Multiply by 2: 2x + 2 = 3x → 2 = x
Then y = 2 + 1 = 3

✔ Solution: (2, 3)

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10)
-7x + 2y = 4
3x - 5y = -2

Use elimination. Multiply first by 5, second by 2:
First: -35x + 10y = 20
Second: 6x - 10y = -4
Add: -29x = 16 → x = -16/29
Now plug into first:
-7*(-16/29) + 2y = 4 → 112/29 + 2y = 4 → 2y = 4 - 112/29 = (116 - 112)/29 = 4/29 → y = 2/29

✔ Solution: (-16/29, 2/29)

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11)
-x + 4y = -4
5x - 8y = 20

Multiply first by 2: -2x + 8y = -8
Add to second:
(-2x + 8y) + (5x - 8y) = -8 + 20 → 3x = 12 → x = 4
Plug into first: -4 + 4y = -4 → 4y = 0 → y = 0

✔ Solution: (4, 0)

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12)
-6x + 5y = 1
6x + 4y = -10

Add them:
(-6x + 5y) + (6x + 4y) = 1 + (-10) → 9y = -9 → y = -1
Plug into second: 6x + 4(-1) = -10 → 6x - 4 = -10 → 6x = -6 → x = -1

✔ Solution: (-1, -1)

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13)
16x - 10y = 10
-8x - 6y = 6

Multiply second by 2: -16x - 12y = 12
Add to first:
(16x - 10y) + (-16x - 12y) = 10 + 12 → -22y = 22 → y = -1
Plug into first: 16x -10(-1)=10 → 16x +10=10 → 16x=0 → x=0

✔ Solution: (0, -1)

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14)
3x - 5y = 17
y = -7

Plug y = -7 into first:
3x -5(-7)=17 → 3x +35=17 → 3x= -18 → x= -6

✔ Solution: (-6, -7)

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15)
5x + 4y = -7
-5x - 2y = 1

Add them:
(5x + 4y) + (-5x - 2y) = -7 + 1 → 2y = -6 → y = -3
Plug into first: 5x +4(-3)= -7 → 5x -12 = -7 → 5x=5 → x=1

✔ Solution: (1, -3)

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16)
-2x + 2y = 6
4x + 2y = -5

Subtract first from second:
(4x + 2y) - (-2x + 2y) = -5 - 6 → 6x = -11 → x = -11/6
Plug into first: -2*(-11/6) + 2y = 6 → 22/6 + 2y = 6 → 11/3 + 2y = 6 → 2y = 6 - 11/3 = 7/3 → y = 7/6

✔ Solution: (-11/6, 7/6)

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Final Answer:

1) (13/7, -5/7)
2) (3, -2)
3) No solution
4) (1, -2)
5) (-12, -7)
6) (-5/3, -11/3)
7) No solution
8) (1, -2)
9) (2, 3)
10) (-16/29, 2/29)
11) (4, 0)
12) (-1, -1)
13) (0, -1)
14) (-6, -7)
15) (1, -3)
16) (-11/6, 7/6)