To solve the given system of equations using elimination, follow these steps:
[tex]\[
\begin{array}{l}
8p + 8q = 19 \quad \text{(1)} \\
5p - 5q = 20 \quad \text{(2)}
\end{array}
\][/tex]
### Step 1: Align the Coefficients for Elimination
Here, we aim to eliminate one of the variables by aligning coefficients. Let's eliminate [tex]\( q \)[/tex].
### Step 2: Equalize the Coefficients of [tex]\( q \)[/tex]
In order to eliminate [tex]\( q \)[/tex], we need to make the coefficients of [tex]\( q \)[/tex] in both equations opposites. For this, we can multiply equation (1) by 5 and equation (2) by 8:
Multiply equation (1) by 5:
[tex]\[
5(8p + 8q) = 5 \cdot 19 \\
40p + 40q = 95 \quad \text{(3)}
\][/tex]
Multiply equation (2) by 8:
[tex]\[
8(5p - 5q) = 8 \cdot 20 \\
40p - 40q = 160 \quad \text{(4)}
\][/tex]
### Step 3: Add or Subtract the Equations to Eliminate [tex]\( q \)[/tex]
Now, add equations (3) and (4) together to eliminate [tex]\( q \)[/tex]:
[tex]\[
(40p + 40q) + (40p - 40q) = 95 + 160 \\
40p + 40p = 255 \\
80p = 255
\][/tex]
### Step 4: Solve for [tex]\( p \)[/tex]
[tex]\[
p = \frac{255}{80} \\
p = 3.1875
\approx 3.19
\][/tex]
### Step 5: Substitute [tex]\( p \)[/tex] Back into One of the Original Equations to Solve for [tex]\( q \)[/tex]
Substitute [tex]\( p = 3.19 \)[/tex] into equation (1):
[tex]\[
8(3.19) + 8q = 19 \\
25.52 + 8q = 19 \\
8q = 19 - 25.52 \\
8q = -6.52 \\
q = \frac{-6.52}{8} \\
q = -0.815
\approx -0.81
\][/tex]
### Conclusion
After solving, we find the solution to be:
[tex]\[
(p, q) = (3.19, -0.81)
\][/tex]
None of the options exactly match our found solution of [tex]\((3.19, -0.81)\)[/tex]. This means the system intersects at [tex]\(\boxed{\text{None of the provided options exactly match the intersection point}}.\)[/tex]
