Systems of Equations worksheets - Free Printable

To solve each system of equations, we will use either the substitution method or the elimination method. Let's go through each problem step by step.

---

Problem 1:

\[
\begin{aligned}
5x + 3y &= -12 \\
5x + 7y &= -8
\end{aligned}
\]

#### Step 1: Eliminate \( x \)
Subtract the first equation from the second:
\[
(5x + 7y) - (5x + 3y) = -8 - (-12)
\]
\[
5x + 7y - 5x - 3y = -8 + 12
\]
\[
4y = 4
\]
\[
y = 1
\]

#### Step 2: Solve for \( x \)
Substitute \( y = 1 \) into the first equation:
\[
5x + 3(1) = -12
\]
\[
5x + 3 = -12
\]
\[
5x = -15
\]
\[
x = -3
\]

#### Solution:
\[
\boxed{x = -3, y = 1}
\]

---

Problem 2:

\[
\begin{aligned}
2x + 5y &= 30 \\
7x + 7y &= 63
\end{aligned}
\]

#### Step 1: Simplify the second equation
Divide the second equation by 7:
\[
x + y = 9
\]

#### Step 2: Solve for \( x \) in terms of \( y \)
From \( x + y = 9 \):
\[
x = 9 - y
\]

#### Step 3: Substitute \( x = 9 - y \) into the first equation
\[
2(9 - y) + 5y = 30
\]
\[
18 - 2y + 5y = 30
\]
\[
18 + 3y = 30
\]
\[
3y = 12
\]
\[
y = 4
\]

#### Step 4: Solve for \( x \)
Substitute \( y = 4 \) into \( x = 9 - y \):
\[
x = 9 - 4
\]
\[
x = 5
\]

#### Solution:
\[
\boxed{x = 5, y = 4}
\]

---

Problem 3:

\[
\begin{aligned}
3x + 6y &= 12 \\
4x + 7y &= 13
\end{aligned}
\]

#### Step 1: Simplify the first equation
Divide the first equation by 3:
\[
x + 2y = 4
\]

#### Step 2: Solve for \( x \) in terms of \( y \)
From \( x + 2y = 4 \):
\[
x = 4 - 2y
\]

#### Step 3: Substitute \( x = 4 - 2y \) into the second equation
\[
4(4 - 2y) + 7y = 13
\]
\[
16 - 8y + 7y = 13
\]
\[
16 - y = 13
\]
\[
-y = -3
\]
\[
y = 3
\]

#### Step 4: Solve for \( x \)
Substitute \( y = 3 \) into \( x = 4 - 2y \):
\[
x = 4 - 2(3)
\]
\[
x = 4 - 6
\]
\[
x = -2
\]

#### Solution:
\[
\boxed{x = -2, y = 3}
\]

---

Problem 4:

\[
\begin{aligned}
5x + 4y &= -27 \\
6x + 7y &= -39
\end{aligned}
\]

#### Step 1: Eliminate \( y \)
Multiply the first equation by 7 and the second equation by 4:
\[
\begin{aligned}
35x + 28y &= -189 \\
24x + 28y &= -156
\end{aligned}
\]

Subtract the second equation from the first:
\[
(35x + 28y) - (24x + 28y) = -189 - (-156)
\]
\[
35x + 28y - 24x - 28y = -189 + 156
\]
\[
11x = -33
\]
\[
x = -3
\]

#### Step 2: Solve for \( y \)
Substitute \( x = -3 \) into the first equation:
\[
5(-3) + 4y = -27
\]
\[
-15 + 4y = -27
\]
\[
4y = -12
\]
\[
y = -3
\]

#### Solution:
\[
\boxed{x = -3, y = -3}
\]

---

Problem 5:

\[
\begin{aligned}
4x + 3y &= -18 \\
3x + 2y &= -13
\end{aligned}
\]

#### Step 1: Eliminate \( y \)
Multiply the first equation by 2 and the second equation by 3:
\[
\begin{aligned}
8x + 6y &= -36 \\
9x + 6y &= -39
\end{aligned}
\]

Subtract the first equation from the second:
\[
(9x + 6y) - (8x + 6y) = -39 - (-36)
\]
\[
9x + 6y - 8x - 6y = -39 + 36
\]
\[
x = -3
\]

#### Step 2: Solve for \( y \)
Substitute \( x = -3 \) into the first equation:
\[
4(-3) + 3y = -18
\]
\[
-12 + 3y = -18
\]
\[
3y = -6
\]
\[
y = -2
\]

#### Solution:
\[
\boxed{x = -3, y = -2}
\]

---

Problem 6:

\[
\begin{aligned}
4x + 7y &= 18 \\
6x + 3y &= 12
\end{aligned}
\]

#### Step 1: Eliminate \( y \)
Multiply the first equation by 3 and the second equation by 7:
\[
\begin{aligned}
12x + 21y &= 54 \\
42x + 21y &= 84
\end{aligned}
\]

Subtract the first equation from the second:
\[
(42x + 21y) - (12x + 21y) = 84 - 54
\]
\[
42x + 21y - 12x - 21y = 30
\]
\[
30x = 30
\]
\[
x = 1
\]

#### Step 2: Solve for \( y \)
Substitute \( x = 1 \) into the first equation:
\[
4(1) + 7y = 18
\]
\[
4 + 7y = 18
\]
\[
7y = 14
\]
\[
y = 2
\]

#### Solution:
\[
\boxed{x = 1, y = 2}
\]

---

Problem 7:

\[
\begin{aligned}
3x + 4y &= 4 \\
6x + 6y &= 6
\end{aligned}
\]

#### Step 1: Simplify the second equation
Divide the second equation by 6:
\[
x + y = 1
\]

#### Step 2: Solve for \( x \) in terms of \( y \)
From \( x + y = 1 \):
\[
x = 1 - y
\]

#### Step 3: Substitute \( x = 1 - y \) into the first equation
\[
3(1 - y) + 4y = 4
\]
\[
3 - 3y + 4y = 4
\]
\[
3 + y = 4
\]
\[
y = 1
\]

#### Step 4: Solve for \( x \)
Substitute \( y = 1 \) into \( x = 1 - y \):
\[
x = 1 - 1
\]
\[
x = 0
\]

#### Solution:
\[
\boxed{x = 0, y = 1}
\]

---

Problem 8:

\[
\begin{aligned}
5x + 6y &= -14 \\
3x + 2y &= -10
\end{aligned}
\]

#### Step 1: Eliminate \( y \)
Multiply the first equation by 1 and the second equation by 3:
\[
\begin{aligned}
5x + 6y &= -14 \\
9x + 6y &= -30
\end{aligned}
\]

Subtract the first equation from the second:
\[
(9x + 6y) - (5x + 6y) = -30 - (-14)
\]
\[
9x + 6y - 5x - 6y = -30 + 14
\]
\[
4x = -16
\]
\[
x = -4
\]

#### Step 2: Solve for \( y \)
Substitute \( x = -4 \) into the first equation:
\[
5(-4) + 6y = -14
\]
\[
-20 + 6y = -14
\]
\[
6y = 6
\]
\[
y = 1
\]

#### Solution:
\[
\boxed{x = -4, y = 1}
\]

---

Problem 9:

\[
\begin{aligned}
2x + 4y &= 14 \\
7x + 4y &= 39
\end{aligned}
\]

#### Step 1: Eliminate \( y \)
Subtract the first equation from the second:
\[
(7x + 4y) - (2x + 4y) = 39 - 14
\]
\[
7x + 4y - 2x - 4y = 25
\]
\[
5x = 25
\]
\[
x = 5
\]

#### Step 2: Solve for \( y \)
Substitute \( x = 5 \) into the first equation:
\[
2(5) + 4y = 14
\]
\[
10 + 4y = 14
\]
\[
4y = 4
\]
\[
y = 1
\]

#### Solution:
\[
\boxed{x = 5, y = 1}
\]

---

Problem 10:

\[
\begin{aligned}
4x + 6y &= 6 \\
7x + 6y &= -3
\end{aligned}
\]

#### Step 1: Eliminate \( y \)
Subtract the first equation from the second:
\[
(7x + 6y) - (4x + 6y) = -3 - 6
\]
\[
7x + 6y - 4x - 6y = -9
\]
\[
3x = -9
\]
\[
x = -3
\]

#### Step 2: Solve for \( y \)
Substitute \( x = -3 \) into the first equation:
\[
4(-3) + 6y = 6
\]
\[
-12 + 6y = 6
\]
\[
6y = 18
\]
\[
y = 3
\]

#### Solution:
\[
\boxed{x = -3, y = 3}
\]

---

Problem 11:

\[
\begin{aligned}
5x + 5y &= 10 \\
3x + 4y &= 8
\end{aligned}
\]

#### Step 1: Simplify the first equation
Divide the first equation by 5:
\[
x + y = 2
\]

#### Step 2: Solve for \( x \) in terms of \( y \)
From \( x + y = 2 \):
\[
x = 2 - y
\]

#### Step 3: Substitute \( x = 2 - y \) into the second equation
\[
3(2 - y) + 4y = 8
\]
\[
6 - 3y + 4y = 8
\]
\[
6 + y = 8
\]
\[
y = 2
\]

#### Step 4: Solve for \( x \)
Substitute \( y = 2 \) into \( x = 2 - y \):
\[
x = 2 - 2
\]
\[
x = 0
\]

#### Solution:
\[
\boxed{x = 0, y = 2}
\]

---

Problem 12:

\[
\begin{aligned}
5x + 4y &= -7 \\
7x + 5y &= -11
\end{aligned}
\]

#### Step 1: Eliminate \( y \)
Multiply the first equation by 5 and the second equation by 4:
\[
\begin{aligned}
25x + 20y &= -35 \\
28x + 20y &= -44
\end{aligned}
\]

Subtract the first equation from the second:
\[
(28x + 20y) - (25x + 20y) = -44 - (-35)
\]
\[
28x + 20y - 25x - 20y = -44 + 35
\]
\[
3x = -9
\]
\[
x = -3
\]

#### Step 2: Solve for \( y \)
Substitute \( x = -3 \) into the first equation:
\[
5(-3) + 4y = -7
\]
\[
-15 + 4y = -7
\]
\[
4y = 8
\]
\[
y = 2
\]

#### Solution:
\[
\boxed{x = -3, y = 2}
\]

---

Problem 13:

\[
\begin{aligned}
5x + 7y &= -41 \\
5x + 5y &= -35
\end{aligned}
\]

#### Step 1: Eliminate \( x \)
Subtract the second equation from the first:
\[
(5x + 7y) - (5x + 5y) = -41 - (-35)
\]
\[
5x + 7y - 5x - 5y = -41 + 35
\]
\[
2y = -6
\]
\[
y = -3
\]

#### Step 2: Solve for \( x \)
Substitute \( y = -3 \) into the first equation:
\[
5x + 7(-3) = -41
\]
\[
5x - 21 = -41
\]
\[
5x = -20
\]
\[
x = -4
\]

#### Solution:
\[
\boxed{x = -4, y = -3}
\]

---

Problem 14:

\[
\begin{aligned}
5x + 3y &= -3 \\
3x + 2y &= -2
\end{aligned}
\]

#### Step 1: Eliminate \( y \)
Multiply the first equation by 2 and the second equation by 3:
\[
\begin{aligned}
10x + 6y &= -6 \\
9x + 6y &= -6
\end{aligned}
\]

Subtract the second equation from the first:
\[
(10x + 6y) - (9x + 6y) = -6 - (-6)
\]
\[
10x + 6y - 9x - 6y = -6 + 6
\]
\[
x = 0
\]

#### Step 2: Solve for \( y \)
Substitute \( x = 0 \) into the first equation:
\[
5(0) + 3y = -3
\]
\[
3y = -3
\]
\[
y = -1
\]

#### Solution:
\[
\boxed{x = 0, y = -1}
\]

---

Problem 15:

\[
\begin{aligned}
4x + 7y &= -22 \\
6x + 4y &= -20
\end{aligned}
\]

#### Step 1: Eliminate \( y \)
Multiply the first equation by 4 and the second equation by 7:
\[
\begin{aligned}
16x + 28y &= -88 \\
42x + 28y &= -140
\end{aligned}
\]

Subtract the first equation from the second:
\[
(42x + 28y) - (16x + 28y) = -140 - (-88)
\]
\[
42x + 28y - 16x - 28y = -140 + 88
\]
\[
26x = -52
\]
\[
x = -2
\]

#### Step 2: Solve for \( y \)
Substitute \( x = -2 \) into the first equation:
\[
4(-2) + 7y = -22
\]
\[
-8 + 7y = -22
\]
\[
7y = -14
\]
\[
y = -2
\]

#### Solution:
\[
\boxed{x = -2, y = -2}
\]

---

Problem 16:

\[
\begin{aligned}
3x + 6y &= 6 \\
4x + 2y &= 8
\end{aligned}
\]

#### Step 1: Simplify the first equation
Divide the first equation by 3:
\[
x + 2y = 2
\]

#### Step 2: Solve for \( x \) in terms of \( y \)
From \( x + 2y = 2 \):
\[
x = 2 - 2y
\]

#### Step 3: Substitute \( x = 2 - 2y \) into the second equation
\[
4(2 - 2y) + 2y = 8
\]
\[
8 - 8y + 2y = 8
\]
\[
8 - 6y = 8
\]
\[
-6y = 0
\]
\[
y = 0
\]

#### Step 4: Solve for \( x \)
Substitute \( y = 0 \) into \( x = 2 - 2y \):
\[
x = 2 - 2(0)
\]
\[
x = 2
\]

#### Solution:
\[
\boxed{x = 2, y = 0}
\]

---

Problem 17:

\[
\begin{aligned}
4x + 2y &= 6 \\
6x + 2y &= 8
\end{aligned}
\]

#### Step 1: Eliminate \( y \)
Subtract the first equation from the second:
\[
(6x + 2y) - (4x + 2y) = 8 - 6
\]
\[
6x + 2y - 4x - 2y = 2
\]
\[
2x = 2
\]
\[
x = 1
\]

#### Step 2: Solve for \( y \)
Substitute \( x = 1 \) into the first equation:
\[
4(1) + 2y = 6
\]
\[
4 + 2y = 6
\]
\[
2y = 2
\]
\[
y = 1
\]

#### Solution:
\[
\boxed{x = 1, y = 1}
\]

---

Problem 18:

\[
\begin{aligned}
3x + 3y &= 21 \\
5x + 2y &= 23
\end{aligned}
\]

#### Step 1: Simplify the first equation
Divide the first equation by 3:
\[
x + y = 7
\]

#### Step 2: Solve for \( x \) in terms of \( y \)
From \( x + y = 7 \):
\[
x = 7 - y
\]

#### Step 3: Substitute \( x = 7 - y \) into the second equation
\[
5(7 - y) + 2y = 23
\]
\[
35 - 5y + 2y = 23
\]
\[
35 - 3y = 23
\]
\[
-3y = -12
\]
\[
y = 4
\]

#### Step 4: Solve for \( x \)
Substitute \( y = 4 \) into \( x = 7 - y \):
\[
x = 7 - 4
\]
\[
x = 3
\]

#### Solution:
\[
\boxed{x = 3, y = 4}
\]

---

Final Answers:

\[
\boxed{
\begin{aligned}
1) & \quad x = -3, y = 1 \\
2) & \quad x = 5, y = 4 \\
3) & \quad x = -2, y = 3 \\
4) & \quad x = -3, y = -3 \\
5) & \quad x = -3, y = -2 \\
6) & \quad x = 1, y = 2 \\
7) & \quad x = 0, y = 1 \\
8) & \quad x = -4, y = 1 \\
9) & \quad x = 5, y = 1 \\
10) & \quad x = -3, y = 3 \\
11) & \quad x = 0, y = 2 \\
12) & \quad x = -3, y = 2 \\
13) & \quad x = -4, y = -3 \\
14) & \quad x = 0, y = -1 \\
15) & \quad x = -2, y = -2 \\
16) & \quad x = 2, y = 0 \\
17) & \quad x = 1, y = 1 \\
18) & \quad x = 3, y = 4 \\
\end{aligned}
}
\]