Sure! Let's solve the system of equations step-by-step.
We are given two equations:
1. [tex]\( 2y - x = 7 \)[/tex]
2. [tex]\( 3y + 2x = 21 \)[/tex]
Step 1: Solving the first equation for [tex]\( x \)[/tex]
From the first equation:
[tex]\[ 2y - x = 7 \][/tex]
We solve for [tex]\( x \)[/tex]:
[tex]\[ x = 2y - 7 \][/tex]
Step 2: Substituting [tex]\( x \)[/tex] in the second equation
Now substitute [tex]\( x = 2y - 7 \)[/tex] into the second equation:
[tex]\[ 3y + 2(2y - 7) = 21 \][/tex]
Simplify inside the parentheses:
[tex]\[ 3y + 4y - 14 = 21 \][/tex]
Combine like terms:
[tex]\[ 7y - 14 = 21 \][/tex]
Step 3: Solving for [tex]\( y \)[/tex]
Add 14 to both sides:
[tex]\[ 7y = 35 \][/tex]
Divide by 7:
[tex]\[ y = 5 \][/tex]
Step 4: Solving for [tex]\( x \)[/tex]
Now substitute [tex]\( y = 5 \)[/tex] back into the equation [tex]\( x = 2y - 7 \)[/tex]:
[tex]\[ x = 2(5) - 7 \][/tex]
[tex]\[ x = 10 - 7 \][/tex]
[tex]\[ x = 3 \][/tex]
Step 5: Writing the solution
So, we have found:
[tex]\[ x = 3 \][/tex]
[tex]\[ y = 5 \][/tex]
This matches option B.
Verification:
It's always good to verify our solution by plugging it back into the original equations.
For the first equation [tex]\( 2y - x = 7 \)[/tex]:
[tex]\[ 2(5) - 3 = 10 - 3 = 7 \][/tex]
For the second equation [tex]\( 3y + 2x = 21 \)[/tex]:
[tex]\[ 3(5) + 2(3) = 15 + 6 = 21 \][/tex]
Both equations are satisfied, confirming our solution is correct.
Thus, the solution to the system of equations is:
[tex]\[
\boxed{\text{B. } x = 3, y = 5}
\][/tex]
