Answer :
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]
[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]
