To solve the system of linear equations:
[tex]\[
\begin{cases}
8p - 3a = 8 \\
2p + 9q = 15
\end{cases}
\][/tex]
we need to find the values of [tex]\( p \)[/tex] and [tex]\( q \)[/tex] in terms of [tex]\( a \)[/tex]. Let's proceed step by step.
1. Solve the first equation for [tex]\( p \)[/tex]:
[tex]\[
8p - 3a = 8
\][/tex]
Isolate [tex]\( p \)[/tex]:
[tex]\[
8p = 3a + 8
\][/tex]
Divide both sides by 8:
[tex]\[
p = \frac{3a}{8} + 1
\][/tex]
2. Substitute [tex]\( p \)[/tex] into the second equation:
The second equation is:
[tex]\[
2p + 9q = 15
\][/tex]
Substitute [tex]\( p = \frac{3a}{8} + 1 \)[/tex]:
[tex]\[
2\left(\frac{3a}{8} + 1\right) + 9q = 15
\][/tex]
3. Simplify the equation:
Distribute the 2:
[tex]\[
2 \cdot \frac{3a}{8} + 2 \cdot 1 + 9q = 15
\][/tex]
[tex]\[
\frac{6a}{8} + 2 + 9q = 15
\][/tex]
Simplify [tex]\(\frac{6a}{8}\)[/tex] to [tex]\(\frac{3a}{4}\)[/tex]:
[tex]\[
\frac{3a}{4} + 2 + 9q = 15
\][/tex]
4. Isolate [tex]\( q \)[/tex]:
Subtract 2 from both sides:
[tex]\[
\frac{3a}{4} + 9q = 13
\][/tex]
Subtract [tex]\(\frac{3a}{4}\)[/tex] from both sides:
[tex]\[
9q = 13 - \frac{3a}{4}
\][/tex]
Divide both sides by 9:
[tex]\[
q = \frac{13}{9} - \frac{3a}{36}
\][/tex]
Simplify [tex]\(\frac{3a}{36}\)[/tex] to [tex]\(\frac{a}{12}\)[/tex]:
[tex]\[
q = \frac{13}{9} - \frac{a}{12}
\][/tex]
Therefore, the solutions for [tex]\( p \)[/tex] and [tex]\( q \)[/tex] in terms of [tex]\( a \)[/tex] are:
[tex]\[
p = \frac{3a}{8} + 1
\][/tex]
[tex]\[
q = \frac{13}{9} - \frac{a}{12}
\][/tex]
These are the values of [tex]\( p \)[/tex] and [tex]\( q \)[/tex] that satisfy the given system of linear equations.
