To solve the system of equations using the substitution method, we will follow these steps:
1. Express [tex]\( x \)[/tex] in terms of [tex]\( y \)[/tex] using the first equation:
[tex]\( x + 2y = 1 \)[/tex]
Solve for [tex]\( x \)[/tex]:
[tex]\[
x = 1 - 2y
\][/tex]
2. Substitute [tex]\( x = 1 - 2y \)[/tex] into the second equation:
The second equation is [tex]\( 5x + 3y = -23 \)[/tex]. Substitute [tex]\( x \)[/tex]:
[tex]\[
5(1 - 2y) + 3y = -23
\][/tex]
3. Simplify and solve for [tex]\( y \)[/tex]:
[tex]\[
5 \cdot 1 - 5 \cdot 2y + 3y = -23
\][/tex]
[tex]\[
5 - 10y + 3y = -23
\][/tex]
[tex]\[
5 - 7y = -23
\][/tex]
Subtract 5 from both sides:
[tex]\[
-7y = -23 - 5
\][/tex]
[tex]\[
-7y = -28
\][/tex]
Divide both sides by -7:
[tex]\[
y = \frac{-28}{-7}
\][/tex]
[tex]\[
y = 4
\][/tex]
4. Substitute [tex]\( y = 4 \)[/tex] back into the expression for [tex]\( x \)[/tex]:
[tex]\[
x = 1 - 2y
\][/tex]
[tex]\[
x = 1 - 2 \cdot 4
\][/tex]
[tex]\[
x = 1 - 8
\][/tex]
[tex]\[
x = -7
\][/tex]
Thus, the solution to the system of equations is:
[tex]\[
(x, y) = (-7, 4)
\][/tex]
