Sure, let's go through the step-by-step solution to solve the given system of linear equations:
[tex]\[
\begin{cases}
3x + y = 7 \\
x + 2y = 9
\end{cases}
\][/tex]
Step 1: Solve one of the equations for one variable.
Let's solve the second equation for [tex]\( x \)[/tex].
[tex]\[
x + 2y = 9 \implies x = 9 - 2y
\][/tex]
Step 2: Substitute the expression for [tex]\( x \)[/tex] into the first equation.
Substitute [tex]\( x = 9 - 2y \)[/tex] into [tex]\( 3x + y = 7 \)[/tex].
[tex]\[
3(9 - 2y) + y = 7
\][/tex]
Step 3: Simplify and solve for [tex]\( y \)[/tex].
[tex]\[
27 - 6y + y = 7 \\
27 - 5y = 7 \\
-5y = 7 - 27 \\
-5y = -20 \\
y = \frac{-20}{-5} \\
y = 4
\][/tex]
Step 4: Substitute [tex]\( y \)[/tex] back into the expression found in Step 1 to solve for [tex]\( x \)[/tex].
Substitute [tex]\( y = 4 \)[/tex] into [tex]\( x = 9 - 2y \)[/tex].
[tex]\[
x = 9 - 2(4) \\
x = 9 - 8 \\
x = 1
\][/tex]
Step 5: Verify the solution by substituting [tex]\( x \)[/tex] and [tex]\( y \)[/tex] back into the original equations.
Check in [tex]\( 3x + y = 7 \)[/tex]:
[tex]\[
3(1) + 4 = 3 + 4 = 7 \quad \text{(which is true)}
\][/tex]
Check in [tex]\( x + 2y = 9 \)[/tex]:
[tex]\[
1 + 2(4) = 1 + 8 = 9 \quad \text{(which is also true)}
\][/tex]
Thus, the solution to the system of equations is:
[tex]\[
x = 1 \quad \text{and} \quad y = 4
\][/tex]
