Practice solving systems of linear equations using the elimination method with these six algebra problems.

Let's solve each system of linear equations using the elimination method. The goal is to eliminate one variable by adding or subtracting the equations, then solve for the remaining variable.

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System 1:

$$
\begin{align*}
(1)\quad & 2x + 3y = -12 \\
(2)\quad & -x - 3y = 18
\end{align*}
$$

Step 1: Add the two equations to eliminate $ y $:

$$
(2x + 3y) + (-x - 3y) = -12 + 18
$$
$$
(2x - x) + (3y - 3y) = 6
$$
$$
x = 6
$$

Step 2: Substitute $ x = 6 $ into equation (1):

$$
2(6) + 3y = -12 \Rightarrow 12 + 3y = -12
\Rightarrow 3y = -24 \Rightarrow y = -8
$$

✔ Solution: $ x = 6, y = -8 $

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System 2:

$$
\begin{align*}
(1)\quad & x + 3y = 18 \\
(2)\quad & -x - 4y = -25
\end{align*}
$$

Step 1: Add the two equations to eliminate $ x $:

$$
(x + 3y) + (-x - 4y) = 18 + (-25)
\Rightarrow (x - x) + (3y - 4y) = -7
\Rightarrow -y = -7 \Rightarrow y = 7
$$

Step 2: Plug $ y = 7 $ into equation (1):

$$
x + 3(7) = 18 \Rightarrow x + 21 = 18 \Rightarrow x = -3
$$

✔ Solution: $ x = -3, y = 7 $

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System 3:

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

Step 1: Add equations to eliminate $ y $:

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

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

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

✔ Solution: $ x = 10, y = -1 $

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System 4:

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

Step 1: Add the equations to eliminate $ x $:

$$
(-6x + 5y) + (6x + 4y) = 1 + (-10)
\Rightarrow 0x + 9y = -9 \Rightarrow y = -1
$$

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

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

✔ Solution: $ x = -1, y = -1 $

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System 5:

$$
\begin{align*}
(1)\quad & 7x - y = -10 \\
(2)\quad & -7x + 5y = -6
\end{align*}
$$

Step 1: Add equations to eliminate $ x $:

$$
(7x - y) + (-7x + 5y) = -10 + (-6)
\Rightarrow 0x + 4y = -16 \Rightarrow y = -4
$$

Step 2: Plug $ y = -4 $ into equation (1):

$$
7x - (-4) = -10 \Rightarrow 7x + 4 = -10 \Rightarrow 7x = -14 \Rightarrow x = -2
$$

✔ Solution: $ x = -2, y = -4 $

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System 6:

$$
\begin{align*}
(1)\quad & -7x - y = 13 \\
(2)\quad & 8x + y = -14
\end{align*}
$$

Step 1: Add equations to eliminate $ y $:

$$
(-7x - y) + (8x + y) = 13 + (-14)
\Rightarrow x = -1
$$

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

$$
8(-1) + y = -14 \Rightarrow -8 + y = -14 \Rightarrow y = -6
$$

✔ Solution: $ x = -1, y = -6 $

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✔ Final Answers:

| System | Solution |
|-------|---------|
| 1 | $ x = 6, y = -8 $ |
| 2 | $ x = -3, y = 7 $ |
| 3 | $ x = 10, y = -1 $ |
| 4 | $ x = -1, y = -1 $ |
| 5 | $ x = -2, y = -4 $ |
| 6 | $ x = -1, y = -6 $ |

Let me know if you'd like a visual explanation or graphing!