A19a - Solving two linear simultaneous equations algebraically by ... - Free Printable

Let's solve each pair of simultaneous equations using the elimination method. The goal is to eliminate one variable by adding or subtracting the equations (after possibly multiplying them by constants), then solving for the remaining variable.

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a)

$$
\begin{align*}
(1)\quad & x + 3y = 26 \\
(2)\quad & x - 3y = -22
\end{align*}
$$

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

$$
(x + 3y) + (x - 3y) = 26 + (-22)
$$
$$
2x = 4 \Rightarrow x = 2
$$

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

$$
2 + 3y = 26 \Rightarrow 3y = 24 \Rightarrow y = 8
$$

✔ Solution: $ x = 2, y = 8 $

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b)

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

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

$$
(4x - 2y) + (3x + 2y) = -12 + 4
$$
$$
7x = -8 \Rightarrow x = -\frac{8}{7}
$$

Step 2: Substitute $ x = -\frac{8}{7} $ into equation (2):

$$
3\left(-\frac{8}{7}\right) + 2y = 4 \Rightarrow -\frac{24}{7} + 2y = 4
$$
$$
2y = 4 + \frac{24}{7} = \frac{28}{7} + \frac{24}{7} = \frac{52}{7}
\Rightarrow y = \frac{26}{7}
$$

✔ Solution: $ x = -\frac{8}{7}, y = \frac{26}{7} $

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c)

$$
\begin{align*}
(1)\quad & 5x - 3y = 6 \\
(2)\quad & -5x + 2y = 1
\end{align*}
$$

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

$$
(5x - 3y) + (-5x + 2y) = 6 + 1
$$
$$
- y = 7 \Rightarrow y = -7
$$

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

$$
5x - 3(-7) = 6 \Rightarrow 5x + 21 = 6 \Rightarrow 5x = -15 \Rightarrow x = -3
$$

✔ Solution: $ x = -3, y = -7 $

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d)

$$
\begin{align*}
(1)\quad & 3x + 2y = 8 \\
(2)\quad & 2x + 2y = 7
\end{align*}
$$

Step 1: Subtract equation (2) from equation (1) to eliminate $ y $:

$$
(3x + 2y) - (2x + 2y) = 8 - 7
$$
$$
x = 1
$$

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

$$
2(1) + 2y = 7 \Rightarrow 2 + 2y = 7 \Rightarrow 2y = 5 \Rightarrow y = \frac{5}{2}
$$

✔ Solution: $ x = 1, y = \frac{5}{2} $

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✔ Final Answers:

a) $ x = 2, y = 8 $
b) $ x = -\frac{8}{7}, y = \frac{26}{7} $
c) $ x = -3, y = -7 $
d) $ x = 1, y = \frac{5}{2} $

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