Math worksheet for solving systems of linear equations using the elimination method.

To solve the systems of linear equations using the elimination method, we will follow these steps:

1. Identify the equations in the system.
2. Manipulate the equations (if necessary) so that one variable can be eliminated by adding or subtracting the equations.
3. Solve for the remaining variable.
4. Substitute the value of the solved variable back into one of the original equations to find the other variable.

Let's solve each system step by step.

---

Problem 1:

\[
\begin{aligned}
1. & \quad -7x - 2y = -36 \\
& \quad -42x + 4y = -152
\end{aligned}
\]

#### Step 1: Eliminate one variable.
To eliminate \( y \), we can multiply the first equation by 2:
\[
2(-7x - 2y) = 2(-36) \implies -14x - 4y = -72
\]
Now we have:
\[
\begin{aligned}
& \quad -14x - 4y = -72 \\
& \quad -42x + 4y = -152
\end{aligned}
\]
Add the two equations:
\[
(-14x - 4y) + (-42x + 4y) = -72 + (-152) \implies -56x = -224
\]
Solve for \( x \):
\[
x = \frac{-224}{-56} = 4
\]

#### Step 2: Substitute \( x = 4 \) back into one of the original equations.
Use the first equation:
\[
-7(4) - 2y = -36 \implies -28 - 2y = -36
\]
Solve for \( y \):
\[
-2y = -36 + 28 \implies -2y = -8 \implies y = 4
\]

#### Solution:
\[
(x, y) = (4, 4)
\]

---

Problem 2:

\[
\begin{aligned}
2. & \quad -x - 6y = 49 \\
& \quad -4x - 2y = 42
\end{aligned}
\]

#### Step 1: Eliminate one variable.
To eliminate \( x \), we can multiply the first equation by 4:
\[
4(-x - 6y) = 4(49) \implies -4x - 24y = 196
\]
Now we have:
\[
\begin{aligned}
& \quad -4x - 24y = 196 \\
& \quad -4x - 2y = 42
\end{aligned}
\]
Subtract the second equation from the first:
\[
(-4x - 24y) - (-4x - 2y) = 196 - 42 \implies -22y = 154
\]
Solve for \( y \):
\[
y = \frac{154}{-22} = -7
\]

#### Step 2: Substitute \( y = -7 \) back into one of the original equations.
Use the first equation:
\[
-x - 6(-7) = 49 \implies -x + 42 = 49
\]
Solve for \( x \):
\[
-x = 49 - 42 \implies -x = 7 \implies x = -7
\]

#### Solution:
\[
(x, y) = (-7, -7)
\]

---

Problem 3:

\[
\begin{aligned}
3. & \quad -4x + 2y = -2 \\
& \quad 20x + 6y = -6
\end{aligned}
\]

#### Step 1: Eliminate one variable.
To eliminate \( y \), we can multiply the first equation by 3 and the second equation by 1:
\[
3(-4x + 2y) = 3(-2) \implies -12x + 6y = -6
\]
Now we have:
\[
\begin{aligned}
& \quad -12x + 6y = -6 \\
& \quad 20x + 6y = -6
\end{aligned}
\]
Subtract the first equation from the second:
\[
(20x + 6y) - (-12x + 6y) = -6 - (-6) \implies 32x = 0
\]
Solve for \( x \):
\[
x = 0
\]

#### Step 2: Substitute \( x = 0 \) back into one of the original equations.
Use the first equation:
\[
-4(0) + 2y = -2 \implies 2y = -2
\]
Solve for \( y \):
\[
y = \frac{-2}{2} = -1
\]

#### Solution:
\[
(x, y) = (0, -1)
\]

---

Problem 4:

\[
\begin{aligned}
4. & \quad -6x - y = 23 \\
& \quad 12x - 6y = -102
\end{aligned}
\]

#### Step 1: Eliminate one variable.
To eliminate \( x \), we can multiply the first equation by 2:
\[
2(-6x - y) = 2(23) \implies -12x - 2y = 46
\]
Now we have:
\[
\begin{aligned}
& \quad -12x - 2y = 46 \\
& \quad 12x - 6y = -102
\end{aligned}
\]
Add the two equations:
\[
(-12x - 2y) + (12x - 6y) = 46 + (-102) \implies -8y = -56
\]
Solve for \( y \):
\[
y = \frac{-56}{-8} = 7
\]

#### Step 2: Substitute \( y = 7 \) back into one of the original equations.
Use the first equation:
\[
-6x - 7 = 23
\]
Solve for \( x \):
\[
-6x = 23 + 7 \implies -6x = 30 \implies x = -5
\]

#### Solution:
\[
(x, y) = (-5, 7)
\]

---

Problem 5:

\[
\begin{aligned}
5. & \quad -2x + 3y = -21 \\
& \quad -6x + 7y = -49
\end{aligned}
\]

#### Step 1: Eliminate one variable.
To eliminate \( x \), we can multiply the first equation by 3:
\[
3(-2x + 3y) = 3(-21) \implies -6x + 9y = -63
\]
Now we have:
\[
\begin{aligned}
& \quad -6x + 9y = -63 \\
& \quad -6x + 7y = -49
\end{aligned}
\]
Subtract the second equation from the first:
\[
(-6x + 9y) - (-6x + 7y) = -63 - (-49) \implies 2y = -14
\]
Solve for \( y \):
\[
y = \frac{-14}{2} = -7
\]

#### Step 2: Substitute \( y = -7 \) back into one of the original equations.
Use the first equation:
\[
-2x + 3(-7) = -21 \implies -2x - 21 = -21
\]
Solve for \( x \):
\[
-2x = -21 + 21 \implies -2x = 0 \implies x = 0
\]

#### Solution:
\[
(x, y) = (0, -7)
\]

---

Problem 6:

\[
\begin{aligned}
6. & \quad 3x + 4y = 8 \\
& \quad -4x - 12y = -44
\end{aligned}
\]

#### Step 1: Eliminate one variable.
To eliminate \( y \), we can multiply the first equation by 3:
\[
3(3x + 4y) = 3(8) \implies 9x + 12y = 24
\]
Now we have:
\[
\begin{aligned}
& \quad 9x + 12y = 24 \\
& \quad -4x - 12y = -44
\end{aligned}
\]
Add the two equations:
\[
(9x + 12y) + (-4x - 12y) = 24 + (-44) \implies 5x = -20
\]
Solve for \( x \):
\[
x = \frac{-20}{5} = -4
\]

#### Step 2: Substitute \( x = -4 \) back into one of the original equations.
Use the first equation:
\[
3(-4) + 4y = 8 \implies -12 + 4y = 8
\]
Solve for \( y \):
\[
4y = 8 + 12 \implies 4y = 20 \implies y = 5
\]

#### Solution:
\[
(x, y) = (-4, 5)
\]

---

Problem 7:

\[
\begin{aligned}
7. & \quad 5x - y = -1 \\
& \quad x - y = 3
\end{aligned}
\]

#### Step 1: Eliminate one variable.
To eliminate \( y \), we can subtract the second equation from the first:
\[
(5x - y) - (x - y) = -1 - 3 \implies 4x = -4
\]
Solve for \( x \):
\[
x = \frac{-4}{4} = -1
\]

#### Step 2: Substitute \( x = -1 \) back into one of the original equations.
Use the second equation:
\[
-1 - y = 3
\]
Solve for \( y \):
\[
-y = 3 + 1 \implies -y = 4 \implies y = -4
\]

#### Solution:
\[
(x, y) = (-1, -4)
\]

---

Problem 8:

\[
\begin{aligned}
8. & \quad -x - 3y = -1 \\
& \quad -4x + 6y = -22
\end{aligned}
\]

#### Step 1: Eliminate one variable.
To eliminate \( y \), we can multiply the first equation by 2:
\[
2(-x - 3y) = 2(-1) \implies -2x - 6y = -2
\]
Now we have:
\[
\begin{aligned}
& \quad -2x - 6y = -2 \\
& \quad -4x + 6y = -22
\end{aligned}
\]
Add the two equations:
\[
(-2x - 6y) + (-4x + 6y) = -2 + (-22) \implies -6x = -24
\]
Solve for \( x \):
\[
x = \frac{-24}{-6} = 4
\]

#### Step 2: Substitute \( x = 4 \) back into one of the original equations.
Use the first equation:
\[
-4 - 3y = -1
\]
Solve for \( y \):
\[
-3y = -1 + 4 \implies -3y = 3 \implies y = -1
\]

#### Solution:
\[
(x, y) = (4, -1)
\]

---

Final Answers:

\[
\boxed{
\begin{aligned}
1. & \quad (4, 4) \\
2. & \quad (-7, -7) \\
3. & \quad (0, -1) \\
4. & \quad (-5, 7) \\
5. & \quad (0, -7) \\
6. & \quad (-4, 5) \\
7. & \quad (-1, -4) \\
8. & \quad (4, -1)
\end{aligned}
}
\]