Sure, let's solve the system of equations using the elimination method step by step:
Given the system of equations:
1. [tex]\(9p + 8q = 2\)[/tex]
2. [tex]\(6p - 5q = 26\)[/tex]
Step 1: Align the coefficients for elimination
- We need to eliminate one of the variables, so let's align the coefficients of [tex]\(p\)[/tex].
To do this, we can multiply the first equation by 6 and the second equation by 9, in order to make the coefficients of [tex]\(p\)[/tex] the same (54):
- First equation: [tex]\(6 \times (9p + 8q) = 6 \times 2 \)[/tex]
- Second equation: [tex]\(9 \times (6p - 5q) = 9 \times 26\)[/tex]
This gives us the new system:
[tex]\[
54p + 48q = 12 \quad \text{(Equation 3)}
\][/tex]
[tex]\[
54p - 45q = 234 \quad \text{(Equation 4)}
\][/tex]
Step 2: Subtract the second equation from the first
Now, subtract Equation 4 from Equation 3 to eliminate [tex]\(p\)[/tex]:
[tex]\[
(54p + 48q) - (54p - 45q) = 12 - 234
\][/tex]
This simplifies to:
[tex]\[
0p + 93q = -222
\][/tex]
or simply:
[tex]\[
93q = -222
\][/tex]
Step 3: Solve for [tex]\(q\)[/tex]
Divide both sides by 93:
[tex]\[
q = \frac{-222}{93} \approx -2.39
\][/tex]
Step 4: Substitute [tex]\(q\)[/tex] back into one of the original equations to solve for [tex]\(p\)[/tex]
Let's use the first equation:
[tex]\[
9p + 8q = 2
\][/tex]
Substitute [tex]\(q = -2.39\)[/tex]:
[tex]\[
9p + 8(-2.39) = 2
\][/tex]
Simplify:
[tex]\[
9p - 19.12 = 2
\][/tex]
Add 19.12 to both sides:
[tex]\[
9p = 21.12
\][/tex]
Divide by 9:
[tex]\[
p = \frac{21.12}{9} \approx 2.34
\][/tex]
So, the solution to the system of equations is:
[tex]\[
p \approx 2.34, \quad q \approx -2.39
\][/tex]
Step 5: Validate the given options with our solution:
a. [tex]\(p = -0.67, q = 2.65\)[/tex] – Wrong
b. [tex]\(p = 2.34, q = -74\)[/tex] – Wrong
c. [tex]\(p = 2.34, q = -2.39\)[/tex] – Correct!
d. [tex]\(p = -1.9, q = -2.39\)[/tex] – Wrong
Thus, the correct solution is option c:
[tex]\[
p = 2.34, q = -2.39
\][/tex]
