To solve the given system of linear equations:
[tex]\[
\begin{cases}
x + 3y = 8 \\
2x - y = 9
\end{cases}
\][/tex]
we'll use the substitution or elimination method. Here is a detailed, step-by-step solution using the elimination method:
1. Write down the equations:
[tex]\[
\begin{cases}
x + 3y = 8 \quad \text{(Equation 1)} \\
2x - y = 9 \quad \text{(Equation 2)}
\end{cases}
\][/tex]
2. Make the coefficients of [tex]\( y \)[/tex] in both equations equal in magnitude:
To eliminate [tex]\( y \)[/tex], we'll scale Equation 2 by 3 so that the coefficient of [tex]\( y \)[/tex] will be the same as for Equation 1:
[tex]\[
3(2x - y) = 3(9)
\][/tex]
This simplifies to:
[tex]\[
6x - 3y = 27 \quad \text{(Equation 3)}
\][/tex]
So now our system of equations looks like this:
[tex]\[
\begin{cases}
x + 3y = 8 \quad \text{(Equation 1)} \\
6x - 3y = 27 \quad \text{(Equation 3)}
\end{cases}
\][/tex]
3. Add the two equations together to eliminate [tex]\( y \)[/tex]:
[tex]\[
(x + 3y) + (6x - 3y) = 8 + 27
\][/tex]
Simplifying this, we get:
[tex]\[
7x = 35
\][/tex]
4. Solve for [tex]\( x \)[/tex]:
[tex]\[
x = \frac{35}{7} = 5
\][/tex]
5. Substitute [tex]\( x = 5 \)[/tex] back into one of the original equations to solve for [tex]\( y \)[/tex]:
Let's use Equation 1:
[tex]\[
x + 3y = 8
\][/tex]
Substitute [tex]\( x = 5 \)[/tex]:
[tex]\[
5 + 3y = 8
\][/tex]
Solving for [tex]\( y \)[/tex]:
[tex]\[
3y = 8 - 5
\][/tex]
[tex]\[
3y = 3
\][/tex]
[tex]\[
y = 1
\][/tex]
6. Summary of the solution:
[tex]\[
x = 5, \quad y = 1
\][/tex]
Thus, the solution to the system of equations is [tex]\((x, y) = (5, 1)\)[/tex].
