Sure! Let's solve this system of linear equations step-by-step.
We have the following system of equations:
[tex]\[
\begin{array}{l}
2x + y = 9 \quad \text{(Equation 1)} \\
x - y = 3 \quad \text{(Equation 2)}
\end{array}
\][/tex]
Step 1: Solve one of the equations for one variable.
Let's solve Equation 2 for [tex]\( x \)[/tex].
[tex]\[
x - y = 3 \implies x = y + 3
\][/tex]
Step 2: Substitute the expression found into the other equation.
Substitute [tex]\( x = y + 3 \)[/tex] into Equation 1:
[tex]\[
2(y + 3) + y = 9
\][/tex]
Step 3: Simplify and solve for [tex]\( y \)[/tex].
Distribute and combine like terms:
[tex]\[
2y + 6 + y = 9
\][/tex]
[tex]\[
3y + 6 = 9
\][/tex]
Subtract 6 from both sides:
[tex]\[
3y = 3
\][/tex]
Divide both sides by 3:
[tex]\[
y = 1
\][/tex]
Step 4: Substitute [tex]\( y \)[/tex] back into the expression found for [tex]\( x \)[/tex].
We have [tex]\( y = 1 \)[/tex] and we previously established [tex]\( x = y + 3 \)[/tex].
[tex]\[
x = 1 + 3
\][/tex]
[tex]\[
x = 4
\][/tex]
Step 5: Write the solution as an ordered pair.
The solution to the system of equations is:
[tex]\[
(x, y) = (4, 1)
\][/tex]
Step 6: Verification.
To ensure our solution is correct, let's substitute [tex]\( x = 4 \)[/tex] and [tex]\( y = 1 \)[/tex] back into the original equations:
1. [tex]\( 2x + y = 9 \)[/tex]:
[tex]\[
2(4) + 1 = 8 + 1 = 9
\][/tex]
2. [tex]\( x - y = 3 \)[/tex]:
[tex]\[
4 - 1 = 3
\][/tex]
Since both equations are satisfied, the solution [tex]\( (x, y) = (4, 1) \)[/tex] is correct.
