To solve the system of simultaneous equations:
[tex]\[
\begin{array}{c}
y - 2x = 8 \\
2x + 5y = 16
\end{array}
\][/tex]
we can use the method of substitution or elimination. Here, we'll demonstrate the substitution method for clarity.
1. Solve the first equation for [tex]\( y \)[/tex]:
[tex]\[ y - 2x = 8 \][/tex]
Add [tex]\( 2x \)[/tex] to both sides:
[tex]\[ y = 2x + 8 \][/tex]
2. Substitute this expression for [tex]\( y \)[/tex] into the second equation:
[tex]\[ 2x + 5(2x + 8) = 16 \][/tex]
3. Expand and simplify the equation:
[tex]\[ 2x + 10x + 40 = 16 \][/tex]
Combine like terms:
[tex]\[ 12x + 40 = 16 \][/tex]
4. Isolate [tex]\( x \)[/tex]:
Subtract 40 from both sides:
[tex]\[ 12x = 16 - 40 \][/tex]
[tex]\[ 12x = -24 \][/tex]
Divide both sides by 12:
[tex]\[ x = -2 \][/tex]
5. Substitute [tex]\( x = -2 \)[/tex] back into the expression for [tex]\( y \)[/tex]:
[tex]\[ y = 2(-2) + 8 \][/tex]
[tex]\[ y = -4 + 8 \][/tex]
[tex]\[ y = 4 \][/tex]
Thus, the solution to the system of equations is:
[tex]\[
\boxed{x = -2 \quad \text{and} \quad y = 4}
\][/tex]
