Sure, let's solve this system of linear equations step-by-step:
We have the following system of equations:
[tex]\[
\begin{cases}
2x + y = 8 \quad \text{(1)} \\
3x - 2y = 5 \quad \text{(2)}
\end{cases}
\][/tex]
Step 1: Solve one of the equations for one variable.
We can solve Equation (1) for [tex]\(y\)[/tex]:
[tex]\[
y = 8 - 2x
\][/tex]
Step 2: Substitute this expression into the other equation.
Substitute [tex]\(y = 8 - 2x\)[/tex] into Equation (2):
[tex]\[
3x - 2(8 - 2x) = 5
\][/tex]
Step 3: Simplify and solve for [tex]\(x\)[/tex].
Distribute the [tex]\(-2\)[/tex] and combine like terms:
[tex]\[
3x - 16 + 4x = 5
\][/tex]
[tex]\[
7x - 16 = 5
\][/tex]
Add 16 to both sides:
[tex]\[
7x = 21
\][/tex]
Divide both sides by 7:
[tex]\[
x = 3
\][/tex]
Step 4: Substitute [tex]\(x\)[/tex] back into the expression for [tex]\(y\)[/tex].
Now we substitute [tex]\(x = 3\)[/tex] back into [tex]\(y = 8 - 2x\)[/tex] to find [tex]\(y\)[/tex]:
[tex]\[
y = 8 - 2(3)
\][/tex]
[tex]\[
y = 8 - 6
\][/tex]
[tex]\[
y = 2
\][/tex]
Final Solution:
Putting it all together, the solution to the system of equations is:
[tex]\[
(x, y) = (3, 2)
\][/tex]
So, the solution is:
[tex]\[
x = 3, \quad y = 2
\][/tex]
