Systems of Equations Elimination worksheet - Free Printable

Let's solve each system of equations by elimination, step by step. The goal is to eliminate one variable by adding or subtracting the equations (after possibly multiplying them by constants), then solve for the remaining variable.

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1)

$$
\begin{align*}
-4x - 2y &= -12 \quad \text{(1)}\\
4x + 8y &= -24 \quad \text{(2)}
\end{align*}
$$

Step: Add equations (1) and (2):
$$
(-4x + 4x) + (-2y + 8y) = -12 + (-24) \\
0x + 6y = -36 \\
6y = -36 \Rightarrow y = -6
$$

Substitute $ y = -6 $ into equation (1):
$$
-4x - 2(-6) = -12 \\
-4x + 12 = -12 \\
-4x = -24 \Rightarrow x = 6
$$

✔ Solution: $ (6, -6) $

---

2)

$$
\begin{align*}
4x + 8y &= 20 \quad \text{(1)}\\
-4x + 2y &= -30 \quad \text{(2)}
\end{align*}
$$

Step: Add equations (1) and (2):
$$
(4x - 4x) + (8y + 2y) = 20 + (-30) \\
0x + 10y = -10 \Rightarrow y = -1
$$

Substitute $ y = -1 $ into equation (1):
$$
4x + 8(-1) = 20 \\
4x - 8 = 20 \\
4x = 28 \Rightarrow x = 7
$$

✔ Solution: $ (7, -1) $

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3)

$$
\begin{align*}
x - y &= 11 \quad \text{(1)}\\
2x + y &= 19 \quad \text{(2)}
\end{align*}
$$

Step: Add equations (1) and (2):
$$
(x + 2x) + (-y + y) = 11 + 19 \\
3x + 0y = 30 \Rightarrow x = 10
$$

Substitute $ x = 10 $ into (1):
$$
10 - y = 11 \Rightarrow -y = 1 \Rightarrow y = -1
$$

✔ Solution: $ (10, -1) $

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4)

$$
\begin{align*}
-6x + 5y &= 1 \quad \text{(1)}\\
6x + 4y &= -10 \quad \text{(2)}
\end{align*}
$$

Step: Add equations (1) and (2):
$$
(-6x + 6x) + (5y + 4y) = 1 + (-10) \\
0x + 9y = -9 \Rightarrow y = -1
$$

Substitute $ y = -1 $ into (1):
$$
-6x + 5(-1) = 1 \\
-6x - 5 = 1 \Rightarrow -6x = 6 \Rightarrow x = -1
$$

✔ Solution: $ (-1, -1) $

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5)

$$
\begin{align*}
-2x - 9y &= -25 \quad \text{(1)}\\
-4x - 9y &= -23 \quad \text{(2)}
\end{align*}
$$

Step: Subtract equation (1) from (2):
$$
(-4x + 2x) + (-9y + 9y) = -23 + 25 \\
-2x + 0y = 2 \Rightarrow x = -1
$$

Substitute $ x = -1 $ into (1):
$$
-2(-1) - 9y = -25 \\
2 - 9y = -25 \Rightarrow -9y = -27 \Rightarrow y = 3
$$

✔ Solution: $ (-1, 3) $

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6)

$$
\begin{align*}
8x + y &= -16 \quad \text{(1)}\\
-3x + y &= -5 \quad \text{(2)}
\end{align*}
$$

Step: Subtract (2) from (1):
$$
(8x + 3x) + (y - y) = -16 + 5 \\
11x = -11 \Rightarrow x = -1
$$

Substitute $ x = -1 $ into (1):
$$
8(-1) + y = -16 \Rightarrow -8 + y = -16 \Rightarrow y = -8
$$

✔ Solution: $ (-1, -8) $

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7)

$$
\begin{align*}
-6x + 6y &= 6 \quad \text{(1)}\\
-6x + 3y &= -12 \quad \text{(2)}
\end{align*}
$$

Step: Subtract (2) from (1):
$$
(-6x + 6x) + (6y - 3y) = 6 + 12 \\
0x + 3y = 18 \Rightarrow y = 6
$$

Substitute $ y = 6 $ into (1):
$$
-6x + 6(6) = 6 \Rightarrow -6x + 36 = 6 \Rightarrow -6x = -30 \Rightarrow x = 5
$$

✔ Solution: $ (5, 6) $

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8)

$$
\begin{align*}
7x + 2y &= 24 \quad \text{(1)}\\
8x + 2y &= 30 \quad \text{(2)}
\end{align*}
$$

Step: Subtract (1) from (2):
$$
(8x - 7x) + (2y - 2y) = 30 - 24 \\
x = 6
$$

Substitute $ x = 6 $ into (1):
$$
7(6) + 2y = 24 \Rightarrow 42 + 2y = 24 \Rightarrow 2y = -18 \Rightarrow y = -9
$$

✔ Solution: $ (6, -9) $

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9)

$$
\begin{align*}
5x + y &= 9 \quad \text{(1)}\\
10x - 7y &= -18 \quad \text{(2)}
\end{align*}
$$

Make coefficients of $ y $ match. Multiply (1) by 7:
$$
35x + 7y = 63 \quad \text{(1a)}
$$

Now add (1a) and (2):
$$
(35x + 10x) + (7y - 7y) = 63 + (-18) \\
45x = 45 \Rightarrow x = 1
$$

Substitute $ x = 1 $ into (1):
$$
5(1) + y = 9 \Rightarrow y = 4
$$

✔ Solution: $ (1, 4) $

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10)

$$
\begin{align*}
-4x + 9y &= 9 \quad \text{(1)}\\
x - 3y &= -6 \quad \text{(2)}
\end{align*}
$$

Multiply (2) by 4:
$$
4x - 12y = -24 \quad \text{(2a)}
$$

Add (1) and (2a):
$$
(-4x + 4x) + (9y - 12y) = 9 + (-24) \\
-3y = -15 \Rightarrow y = 5
$$

Substitute $ y = 5 $ into (2):
$$
x - 3(5) = -6 \Rightarrow x - 15 = -6 \Rightarrow x = 9
$$

✔ Solution: $ (9, 5) $

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11)

$$
\begin{align*}
-3x + 7y &= -16 \quad \text{(1)}\\
-9x + 5y &= 16 \quad \text{(2)}
\end{align*}
$$

Multiply (1) by 3:
$$
-9x + 21y = -48 \quad \text{(1a)}
$$

Now subtract (2) from (1a):
$$
(-9x + 9x) + (21y - 5y) = -48 - 16 \\
16y = -64 \Rightarrow y = -4
$$

Substitute $ y = -4 $ into (1):
$$
-3x + 7(-4) = -16 \Rightarrow -3x - 28 = -16 \Rightarrow -3x = 12 \Rightarrow x = -4
$$

✔ Solution: $ (-4, -4) $

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12)

$$
\begin{align*}
-7x + y &= -19 \quad \text{(1)}\\
-2x + 3y &= -19 \quad \text{(2)}
\end{align*}
$$

Multiply (1) by 3:
$$
-21x + 3y = -57 \quad \text{(1a)}
$$

Subtract (2) from (1a):
$$
(-21x + 2x) + (3y - 3y) = -57 + 19 \\
-19x = -38 \Rightarrow x = 2
$$

Substitute $ x = 2 $ into (1):
$$
-7(2) + y = -19 \Rightarrow -14 + y = -19 \Rightarrow y = -5
$$

✔ Solution: $ (2, -5) $

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✔ Final Answers:

| Problem | Solution |
|--------|----------|
| 1 | $ (6, -6) $ |
| 2 | $ (7, -1) $ |
| 3 | $ (10, -1) $ |
| 4 | $ (-1, -1) $ |
| 5 | $ (-1, 3) $ |
| 6 | $ (-1, -8) $ |
| 7 | $ (5, 6) $ |
| 8 | $ (6, -9) $ |
| 9 | $ (1, 4) $ |
| 10 | $ (9, 5) $ |
| 11 | $ (-4, -4) $ |
| 12 | $ (2, -5) $ |

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