To solve the system of equations by substitution, we will start by isolating one variable in one of the equations and then substituting it in the other equation.
Given the system:
[tex]\[
\begin{array}{l}
0.5x + 0.25y = 36 \\
y + 18 = 16x
\end{array}
\][/tex]
1. Isolate [tex]\( y \)[/tex] in the second equation:
[tex]\[
y + 18 = 16x
\][/tex]
Subtract 18 from both sides:
[tex]\[
y = 16x - 18
\][/tex]
2. Substitute [tex]\( y = 16x - 18 \)[/tex] into the first equation:
[tex]\[
0.5x + 0.25(16x - 18) = 36
\][/tex]
Distribute [tex]\( 0.25 \)[/tex] in the equation:
[tex]\[
0.5x + 4x - 4.5 = 36
\][/tex]
Combine like terms:
[tex]\[
4.5x - 4.5 = 36
\][/tex]
Add 4.5 to both sides:
[tex]\[
4.5x = 40.5
\][/tex]
Solve for [tex]\( x \)[/tex]:
[tex]\[
x = \frac{40.5}{4.5} = 9
\][/tex]
3. Use [tex]\( x = 9 \)[/tex] to find [tex]\( y \)[/tex] using the isolated expression [tex]\( y = 16x - 18 \)[/tex]:
[tex]\[
y = 16(9) - 18
\][/tex]
Calculate the value:
[tex]\[
y = 144 - 18 = 126
\][/tex]
So, the solution of the system is [tex]\((x, y) = (9, 126)\)[/tex].
4. Check the solution:
Substitute [tex]\( x = 9 \)[/tex] and [tex]\( y = 126 \)[/tex] back into the original equations to verify:
First equation:
[tex]\[
0.5(9) + 0.25(126) = 4.5 + 31.5 = 36
\][/tex]
True.
Second equation:
[tex]\[
126 + 18 = 16(9) \\
144 = 144
\][/tex]
True.
Since both equations are satisfied, the solution [tex]\((9, 126)\)[/tex] is correct.
So, the correct answer is [tex]\((9, 126)\)[/tex], which corresponds to option [tex]\((b)\)[/tex].
