Linear Equations Worksheets with Answer Key - Free Printable

Let's solve each pair of linear equations using the substitution method. The idea is to solve one equation for one variable and substitute that expression into the other equation.

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Problem 1:

$$
\begin{align*}
(1)\quad & -5x + y = -7 \\
(2)\quad & -3x - 2y = -12
\end{align*}
$$

Step 1: Solve equation (1) for $ y $:
$$
y = 5x - 7
$$

Step 2: Substitute into equation (2):
$$
-3x - 2(5x - 7) = -12
$$
$$
-3x - 10x + 14 = -12
$$
$$
-13x + 14 = -12
$$
$$
-13x = -26 \Rightarrow x = 2
$$

Step 3: Plug back into $ y = 5x - 7 $:
$$
y = 5(2) - 7 = 10 - 7 = 3
$$

✔ Solution: $ (x, y) = (2, 3) $

---

Problem 2:

$$
\begin{align*}
(1)\quad & -2x + 6y = 6 \\
(2)\quad & -7x + 8y = -5
\end{align*}
$$

Step 1: Solve equation (1) for $ x $:
$$
-2x = 6 - 6y \Rightarrow x = -3 + 3y
$$

Step 2: Substitute into equation (2):
$$
-7(-3 + 3y) + 8y = -5
$$
$$
21 - 21y + 8y = -5
$$
$$
21 - 13y = -5
$$
$$
-13y = -26 \Rightarrow y = 2
$$

Step 3: Plug back into $ x = -3 + 3y $:
$$
x = -3 + 3(2) = -3 + 6 = 3
$$

✔ Solution: $ (x, y) = (3, 2) $

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Problem 3:

$$
\begin{align*}
(1)\quad & -5x - y = 21 \\
(2)\quad & -4x + y = 6
\end{align*}
$$

Step 1: Solve equation (2) for $ y $:
$$
y = 4x + 6
$$

Step 2: Substitute into equation (1):
$$
-5x - (4x + 6) = 21
$$
$$
-5x - 4x - 6 = 21
$$
$$
-9x = 27 \Rightarrow x = -3
$$

Step 3: Plug into $ y = 4x + 6 $:
$$
y = 4(-3) + 6 = -12 + 6 = -6
$$

✔ Solution: $ (x, y) = (-3, -6) $

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Problem 4:

$$
\begin{align*}
(1)\quad & y = -3x \\
(2)\quad & 4x - 2y = -20
\end{align*}
$$

Step 1: Equation (1) already gives $ y = -3x $

Step 2: Substitute into (2):
$$
4x - 2(-3x) = -20
$$
$$
4x + 6x = -20 \Rightarrow 10x = -20 \Rightarrow x = -2
$$

Step 3: Plug into $ y = -3x $:
$$
y = -3(-2) = 6
$$

✔ Solution: $ (x, y) = (-2, 6) $

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Problem 5:

$$
\begin{align*}
(1)\quad & x = 3y + 1 \\
(2)\quad & 2x + 4y = 12
\end{align*}
$$

Step 1: Equation (1) already gives $ x = 3y + 1 $

Step 2: Substitute into (2):
$$
2(3y + 1) + 4y = 12
$$
$$
6y + 2 + 4y = 12 \Rightarrow 10y = 10 \Rightarrow y = 1
$$

Step 3: Plug into $ x = 3y + 1 $:
$$
x = 3(1) + 1 = 4
$$

✔ Solution: $ (x, y) = (4, 1) $

---

Problem 6:

$$
\begin{align*}
(1)\quad & -5x - 8y = 17 \\
(2)\quad & 2x - 7y = -17
\end{align*}
$$

Step 1: Solve equation (2) for $ x $:
$$
2x = 7y - 17 \Rightarrow x = \frac{7y - 17}{2}
$$

Step 2: Substitute into equation (1):
$$
-5\left(\frac{7y - 17}{2}\right) - 8y = 17
$$
Multiply both sides by 2 to eliminate denominator:
$$
-5(7y - 17) - 16y = 34
$$
$$
-35y + 85 - 16y = 34
$$
$$
-51y = -51 \Rightarrow y = 1
$$

Step 3: Plug into $ x = \frac{7(1) - 17}{2} = \frac{-10}{2} = -5 $

✔ Solution: $ (x, y) = (-5, 1) $

---

Problem 7:

$$
\begin{align*}
(1)\quad & x + 9y = -1 \\
(2)\quad & 2x + 4y = 5
\end{align*}
$$

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

Step 2: Substitute into (2):
$$
2(-1 - 9y) + 4y = 5
$$
$$
-2 - 18y + 4y = 5
$$
$$
-2 - 14y = 5 \Rightarrow -14y = 7 \Rightarrow y = -\frac{1}{2}
$$

Step 3: Plug into $ x = -1 - 9y $:
$$
x = -1 - 9(-\frac{1}{2}) = -1 + \frac{9}{2} = \frac{7}{2}
$$

✔ Solution: $ (x, y) = \left(\frac{7}{2}, -\frac{1}{2}\right) $

---

Problem 8:

$$
\begin{align*}
(1)\quad & x - y = 11 \\
(2)\quad & 3x + 10y = -6
\end{align*}
$$

Step 1: Solve equation (1) for $ x $:
$$
x = y + 11
$$

Step 2: Substitute into (2):
$$
3(y + 11) + 10y = -6
$$
$$
3y + 33 + 10y = -6 \Rightarrow 13y = -39 \Rightarrow y = -3
$$

Step 3: Plug into $ x = y + 11 $:
$$
x = -3 + 11 = 8
$$

✔ Solution: $ (x, y) = (8, -3) $

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✔ Final Answers:

| Problem | Solution |
|--------|----------|
| 1 | $ (2, 3) $ |
| 2 | $ (3, 2) $ |
| 3 | $ (-3, -6) $ |
| 4 | $ (-2, 6) $ |
| 5 | $ (4, 1) $ |
| 6 | $ (-5, 1) $ |
| 7 | $ \left(\frac{7}{2}, -\frac{1}{2}\right) $ |
| 8 | $ (8, -3) $ |

Let me know if you'd like these checked or graphed!