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 1: Add the two equations to eliminate $x$:
$$
(-4x - 2y) + (4x + 8y) = -12 + (-24)
$$
$$
6y = -36 \Rightarrow y = -6
$$

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

✔ Solution: $(6, -6)$

---

2)

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

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

Step 2: Plug $y = -1$ into (1):
$$
4x + 8(-1) = 20 \Rightarrow 4x - 8 = 20 \Rightarrow 4x = 28 \Rightarrow x = 7
$$

✔ 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 1: Add equations to eliminate $y$:
$$
(x - y) + (2x + y) = 11 + 19 \Rightarrow 3x = 30 \Rightarrow x = 10
$$

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

Step 2: Plug $y = -1$ into (1):
$$
-6x + 5(-1) = 1 \Rightarrow -6x - 5 = 1 \Rightarrow -6x = 6 \Rightarrow x = -1
$$

✔ 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 1: Subtract (1) from (2) to eliminate $y$:
$$
(-4x - 9y) - (-2x - 9y) = -23 - (-25)
\Rightarrow -4x - 9y + 2x + 9y = 2 \Rightarrow -2x = 2 \Rightarrow x = -1
$$

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

✔ Solution: $(-1, 3)$

---

6)

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

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

Step 2: Plug $x = -1$ into (1):
$$
8(-1) + y = -16 \Rightarrow -8 + y = -16 \Rightarrow y = -8
$$

✔ Solution: $(-1, -8)$

---

7)

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

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

Step 2: Plug $y = 6$ into (1):
$$
-6x + 6(6) = 6 \Rightarrow -6x + 36 = 6 \Rightarrow -6x = -30 \Rightarrow x = 5
$$

✔ Solution: $(5, 6)$

---

8)

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

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

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

✔ Solution: $(6, -9)$

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9)

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

Step 1: Multiply (1) by 7:
$$
7(5x + y) = 7(9) \Rightarrow 35x + 7y = 63 \quad \text{(3)}
$$

Now add (3) and (2):
$$
(35x + 7y) + (10x - 7y) = 63 + (-18) \Rightarrow 45x = 45 \Rightarrow x = 1
$$

Step 2: Plug $x = 1$ into (1):
$$
5(1) + y = 9 \Rightarrow y = 4
$$

✔ Solution: $(1, 4)$

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10)

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

Step 1: Multiply (2) by 4:
$$
4(x - 3y) = 4(-6) \Rightarrow 4x - 12y = -24 \quad \text{(3)}
$$

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

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

✔ Solution: $(9, 5)$

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11)

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

Step 1: Multiply (1) by 3:
$$
3(-3x + 7y) = 3(-16) \Rightarrow -9x + 21y = -48 \quad \text{(3)}
$$

Now subtract (2) from (3):
$$
(-9x + 21y) - (-9x + 5y) = -48 - 16
\Rightarrow -9x + 21y + 9x - 5y = -64 \Rightarrow 16y = -64 \Rightarrow y = -4
$$

Step 2: Plug $y = -4$ into (1):
$$
-3x + 7(-4) = -16 \Rightarrow -3x - 28 = -16 \Rightarrow -3x = 12 \Rightarrow x = -4
$$

✔ Solution: $(-4, -4)$

---

12)

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

Step 1: Multiply (1) by 3:
$$
3(-7x + y) = 3(-19) \Rightarrow -21x + 3y = -57 \quad \text{(3)}
$$

Now subtract (2) from (3):
$$
(-21x + 3y) - (-2x + 3y) = -57 - (-19)
\Rightarrow -21x + 3y + 2x - 3y = -38 \Rightarrow -19x = -38 \Rightarrow x = 2
$$

Step 2: Plug $x = 2$ 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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