Systems of Equations Elimination Method Task Card Self Checking ... - Free Printable

Let's solve each of the 10 systems of linear equations step by step. We'll use either substitution, elimination, or matrix methods as appropriate.

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Question 1

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

Step 1: Add equations (1) and (2) to eliminate $x$:

$$
(-4x + 2y) + (4x + 8y) = -12 + (-24)
\Rightarrow 10y = -36 \Rightarrow y = -3.6
$$

Step 2: Plug $y = -3.6$ into equation (1):

$$
-4x + 2(-3.6) = -12 \Rightarrow -4x - 7.2 = -12 \Rightarrow -4x = -4.8 \Rightarrow x = 1.2
$$

✔ Answer: $x = 1.2$, $y = -3.6$

---

Question 2

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

Step 1: Add equations (1) and (2):

$$
(4x + 8y) + (-4x + 2y) = 20 + (-30) \Rightarrow 10y = -10 \Rightarrow y = -1
$$

Step 2: Plug $y = -1$ into (1):

$$
4x + 8(-1) = 20 \Rightarrow 4x - 8 = 20 \Rightarrow 4x = 28 \Rightarrow x = 7
$$

✔ Answer: $x = 7$, $y = -1$

---

Question 3

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

Step 1: Add equations (1) and (2):

$$
(x - y) + (2x + y) = 11 + 19 \Rightarrow 3x = 30 \Rightarrow x = 10
$$

Step 2: Plug $x = 10$ into (1):

$$
10 - y = 11 \Rightarrow y = -1
$$

✔ Answer: $x = 10$, $y = -1$

---

Question 4

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

Step 1: Add equations (1) and (2):

$$
(-6x + 5y) + (6x + 4y) = 1 + (-10) \Rightarrow 9y = -9 \Rightarrow y = -1
$$

Step 2: Plug $y = -1$ into (1):

$$
-6x + 5(-1) = 1 \Rightarrow -6x - 5 = 1 \Rightarrow -6x = 6 \Rightarrow x = -1
$$

✔ Answer: $x = -1$, $y = -1$

---

Question 5

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

Step 1: Subtract (1) from (2):

$$
(4x - 9y) - (-2x - 9y) = -23 - (-25)
\Rightarrow 4x - 9y + 2x + 9y = 2 \Rightarrow 6x = 2 \Rightarrow x = \frac{1}{3}
$$

Step 2: Plug $x = \frac{1}{3}$ into (1):

$$
-2(\frac{1}{3}) - 9y = -25 \Rightarrow -\frac{2}{3} - 9y = -25
\Rightarrow -9y = -25 + \frac{2}{3} = -\frac{75}{3} + \frac{2}{3} = -\frac{73}{3}
\Rightarrow y = \frac{73}{27}
$$

✔ Answer: $x = \frac{1}{3}$, $y = \frac{73}{27}$

---

Question 6

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

Step 1: Subtract (2) from (1):

$$
(8x + y) - (-3x + y) = -16 - 5 \Rightarrow 8x + y + 3x - y = -21 \Rightarrow 11x = -21 \Rightarrow x = -\frac{21}{11}
$$

Step 2: Plug $x = -\frac{21}{11}$ into (2):

$$
-3(-\frac{21}{11}) + y = 5 \Rightarrow \frac{63}{11} + y = 5 \Rightarrow y = 5 - \frac{63}{11} = \frac{55}{11} - \frac{63}{11} = -\frac{8}{11}
$$

✔ Answer: $x = -\frac{21}{11}$, $y = -\frac{8}{11}$

---

Question 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):

$$
-6x + 6(6) = 6 \Rightarrow -6x + 36 = 6 \Rightarrow -6x = -30 \Rightarrow x = 5
$$

✔ Answer: $x = 5$, $y = 6$

---

Question 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):

$$
7(6) + 2y = 24 \Rightarrow 42 + 2y = 24 \Rightarrow 2y = -18 \Rightarrow y = -9
$$

✔ Answer: $x = 6$, $y = -9$

---

Question 9

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

Step 1: Solve (1) for $y$:
$y = 9 - 5x$

Step 2: Substitute into (2):

$$
10x - 7(9 - 5x) = -18 \Rightarrow 10x - 63 + 35x = -18 \Rightarrow 45x = 45 \Rightarrow x = 1
$$

Step 3: Plug $x = 1$ into $y = 9 - 5x$:
$y = 9 - 5(1) = 4$

✔ Answer: $x = 1$, $y = 4$

---

Question 10

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

Step 1: Solve (2) for $x$:
$x = 3y - 6$

Step 2: Substitute into (1):

$$
-4(3y - 6) + 9y = 9 \Rightarrow -12y + 24 + 9y = 9 \Rightarrow -3y = -15 \Rightarrow y = 5
$$

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

✔ Answer: $x = 9$, $y = 5$

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✔ Final Answers Summary:

| Question | $x$ | $y$ |
|--------|-----|-----|
| 1 | 1.2 | -3.6 |
| 2 | 7 | -1 |
| 3 | 10 | -1 |
| 4 | -1 | -1 |
| 5 | $\frac{1}{3}$ | $\frac{73}{27}$ |
| 6 | $-\frac{21}{11}$ | $-\frac{8}{11}$ |
| 7 | 5 | 6 |
| 8 | 6 | -9 |
| 9 | 1 | 4 |
| 10 | 9 | 5 |

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