To find the solution to the system of equations given by:
[tex]\[
\begin{aligned}
&1.\; x - 2y = -23 \\
&2.\; x - y = 7 \\
\end{aligned}
\][/tex]
we can use the method of substitution or elimination. Here is a step-by-step method to solve it using elimination:
1. Step 1: Align the equations for elimination
[tex]\[
\begin{aligned}
&1.\; x - 2y = -23 \quad \text{(Equation 1)} \\
&2.\; x - y = 7 \quad \text{(Equation 2)}
\end{aligned}
\][/tex]
2. Step 2: Eliminate one variable
To eliminate the variable [tex]\( x \)[/tex], we subtract Equation 2 from Equation 1:
[tex]\[
(x - 2y) - (x - y) = -23 - 7
\][/tex]
Simplifying this, we get:
[tex]\[
x - 2y - x + y = -23 - 7
\][/tex]
[tex]\[
-y = -30
\][/tex]
[tex]\[
y = 30
\][/tex]
3. Step 3: Substitute the value of [tex]\( y \)[/tex] back into one of the original equations
We can use Equation 2 for substitution:
[tex]\[
x - y = 7
\][/tex]
Substituting [tex]\( y = 30 \)[/tex]:
[tex]\[
x - 30 = 7
\][/tex]
Solving for [tex]\( x \)[/tex]:
[tex]\[
x = 37
\][/tex]
4. Step 4: Verify the solution
Use the values [tex]\( x = 37 \)[/tex] and [tex]\( y = 30 \)[/tex] to check both original equations:
[tex]\[
\text{Equation 1: } 37 - 2(30) = 37 - 60 = -23 \quad \text{(Correct)}
\][/tex]
[tex]\[
\text{Equation 2: } 37 - 30 = 7 \quad \text{(Correct)}
\][/tex]
Thus, the solution to the system of equations is:
[tex]\[
\boxed{37, \ 30}
\][/tex]
