To solve the system of simultaneous equations given:
[tex]\[
\begin{cases}
2p - 3q = 4 \\
3p + 2q = 19
\end{cases}
\][/tex]
we will use the method of elimination to find the values of \( p \) and \( q \).
### Step 1: Eliminate one variable
First, let's eliminate \( q \). To do this, we need to make the coefficients of \( q \) in both equations equal. We'll do this by multiplying each equation by a number that will make the coefficients of \( q \) equal when added or subtracted.
For the first equation \( 2p - 3q = 4 \), multiply by 2:
[tex]\[
4p - 6q = 8
\][/tex]
For the second equation \( 3p + 2q = 19 \), multiply by 3:
[tex]\[
9p + 6q = 57
\][/tex]
Now we have the system:
[tex]\[
\begin{cases}
4p - 6q = 8 \\
9p + 6q = 57
\end{cases}
\][/tex]
### Step 2: Add or subtract the equations
Next, add the two equations to eliminate \( q \):
[tex]\[
(4p - 6q) + (9p + 6q) = 8 + 57
\][/tex]
[tex]\[
4p + 9p = 65
\][/tex]
[tex]\[
13p = 65
\][/tex]
### Step 3: Solve for \( p \)
Divide both sides by 13:
[tex]\[
p = \frac{65}{13}
\][/tex]
[tex]\[
p = 5
\][/tex]
### Step 4: Substitute \( p \) back into one of the original equations to solve for \( q \)
We'll substitute \( p = 5 \) into the first equation \( 2p - 3q = 4 \):
[tex]\[
2(5) - 3q = 4
\][/tex]
[tex]\[
10 - 3q = 4
\][/tex]
Subtract 10 from both sides:
[tex]\[
-3q = 4 - 10
\][/tex]
[tex]\[
-3q = -6
\][/tex]
Divide both sides by -3:
[tex]\[
q = \frac{-6}{-3}
\][/tex]
[tex]\[
q = 2
\][/tex]
### Solution
The solution to the system of equations is:
[tex]\[
(p, q) = (5, 2)
\][/tex]
