Linear Equations Worksheets with Answer Key - Free Printable

Let's solve each pair of linear equations using the substitution method. The goal 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):
$$
-3x - 2(5x - 7) = -12 \\
-3x - 10x + 14 = -12 \\
-13x + 14 = -12 \\
-13x = -26 \\
x = 2
$$

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

✔ Solution: $ x = 2, y = 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):
$$
-7(-3 + 3y) + 8y = -5 \\
21 - 21y + 8y = -5 \\
21 - 13y = -5 \\
-13y = -26 \\
y = 2
$$

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

✔ Solution: $ x = 3, y = 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):
$$
-5x - (4x + 6) = 21 \\
-5x - 4x - 6 = 21 \\
-9x = 27 \\
x = -3
$$

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

✔ Solution: $ x = -3, y = -6 $

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Problem 4:

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

Step 1: Already solved for $ y $. Substitute into (2):
$$
4x - 2(-3x) = -20 \\
4x + 6x = -20 \\
10x = -20 \\
x = -2
$$

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

✔ Solution: $ x = -2, y = 6 $

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Problem 5:

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

Step 1: Already solved for $ x $. Substitute into (2):
$$
2(3y + 1) + 4y = 12 \\
6y + 2 + 4y = 12 \\
10y + 2 = 12 \\
10y = 10 \\
y = 1
$$

Step 2: Plug into $ x = 3y + 1 $:
$$
x = 3(1) + 1 = 4
$$

✔ Solution: $ x = 4, y = 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):
$$
-5\left(\frac{7y - 17}{2}\right) - 8y = 17 \\
\frac{-35y + 85}{2} - 8y = 17
$$

Multiply entire equation by 2 to eliminate denominator:
$$
-35y + 85 - 16y = 34 \\
-51y + 85 = 34 \\
-51y = -51 \\
y = 1
$$

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

✔ Solution: $ x = -5, y = 1 $

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

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

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

---

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):
$$
3(y + 11) + 10y = -6 \\
3y + 33 + 10y = -6 \\
13y + 33 = -6 \\
13y = -39 \\
y = -3
$$

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

✔ Solution: $ x = 8, y = -3 $

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✔ Final Answers:

| Problem | Solution |
|--------|----------|
| 1 | $ x = 2, y = 3 $ |
| 2 | $ x = 3, y = 2 $ |
| 3 | $ x = -3, y = -6 $ |
| 4 | $ x = -2, y = 6 $ |
| 5 | $ x = 4, y = 1 $ |
| 6 | $ x = -5, y = 1 $ |
| 7 | $ x = \frac{7}{2}, y = -\frac{1}{2} $ |
| 8 | $ x = 8, y = -3 $ |

Let me know if you'd like these checked with graphing or elimination methods!