Sure, let's solve this system of equations step-by-step.
The given system of equations is:
[tex]\[
\begin{cases}
y - 2x = 8 \\
2x + 5y = 16
\end{cases}
\][/tex]
Step 1: Solve the first equation for \( y \):
[tex]\[
y - 2x = 8
\][/tex]
Add \( 2x \) to both sides:
[tex]\[
y = 2x + 8
\][/tex]
Step 2: Substitute \( y = 2x + 8 \) into the second equation:
The second equation is:
[tex]\[
2x + 5y = 16
\][/tex]
Substitute \( y \) from step 1:
[tex]\[
2x + 5(2x + 8) = 16
\][/tex]
Step 3: Expand and simplify:
[tex]\[
2x + 10x + 40 = 16
\][/tex]
Combine like terms:
[tex]\[
12x + 40 = 16
\][/tex]
Step 4: Solve for \( x \):
Subtract 40 from both sides:
[tex]\[
12x = 16 - 40
\][/tex]
[tex]\[
12x = -24
\][/tex]
Divide both sides by 12:
[tex]\[
x = -2
\][/tex]
Step 5: Substitute \( x = -2 \) back into the expression for \( y \):
We already have \( y = 2x + 8 \):
[tex]\[
y = 2(-2) + 8
\][/tex]
[tex]\[
y = -4 + 8
\][/tex]
[tex]\[
y = 4
\][/tex]
Solution:
The solution to the system of equations is:
[tex]\[
x = -2 \quad \text{and} \quad y = 4
\][/tex]
So the coordinates [tex]\((x, y)\)[/tex] that satisfy both equations are [tex]\((-2, 4)\)[/tex].
