To solve for the values of [tex]\( x \)[/tex] and [tex]\( y \)[/tex] given the system of equations:
[tex]\[
\begin{aligned}
1. & \quad 7y - 5x = 45 \\
2. & \quad x = 8 - 2y
\end{aligned}
\][/tex]
we will follow these steps:
1. Express [tex]\( y \)[/tex] in terms of [tex]\( x \)[/tex] from the second equation:
Start with the equation:
[tex]\[
x = 8 - 2y
\][/tex]
Rearrange to solve for [tex]\( y \)[/tex]:
[tex]\[
2y = 8 - x \\
y = \frac{8 - x}{2}
\][/tex]
2. Substitute this expression for [tex]\( y \)[/tex] into the first equation:
Original first equation:
[tex]\[
7y - 5x = 45
\][/tex]
Substitute [tex]\( y = \frac{8 - x}{2} \)[/tex]:
[tex]\[
7 \left(\frac{8 - x}{2}\right) - 5x = 45
\][/tex]
3. Simplify and solve for [tex]\( x \)[/tex]:
Distribute and simplify:
[tex]\[
\frac{7(8 - x)}{2} - 5x = 45
\][/tex]
[tex]\[
\frac{56 - 7x}{2} - 5x = 45
\][/tex]
Multiply through by 2 to clear the fraction:
[tex]\[
56 - 7x - 10x = 90
\][/tex]
[tex]\[
56 - 17x = 90
\][/tex]
Subtract 56 from both sides:
[tex]\[
-17x = 34
\][/tex]
Divide by -17:
[tex]\[
x = \frac{34}{-17}
\][/tex]
[tex]\[
x = -2
\][/tex]
4. Substitute [tex]\( x \)[/tex] back into the expression for [tex]\( y \)[/tex]:
Use the expression [tex]\( y = \frac{8 - x}{2} \)[/tex]:
[tex]\[
y = \frac{8 - (-2)}{2}
\][/tex]
[tex]\[
y = \frac{8 + 2}{2}
\][/tex]
[tex]\[
y = \frac{10}{2}
\][/tex]
[tex]\[
y = 5
\][/tex]
Therefore, the solution to the given system of equations is:
[tex]\[
x = -2 \quad \text{and} \quad y = 5
\][/tex]
None of the given options ([tex]\( x = 5, y = 10 \)[/tex] or [tex]\( x = 4, y = 2 \)[/tex]) match this solution. So, neither option is correct.
