Question

The system of equations is: \[ 7x + 3y = 12 \\ 3x + 7y = 6 \] If \(x\) and \(y\) satisfy the system of equations, what is the value of \(x - y\)?

• A. \( \frac{2}{3} \)
• B. \( \frac{3}{2} \)
• C. 1
• D. 4
• E. 6

Hint:

When solving systems of equations, try using elimination or substitution to simplify and find the value of variables.

Answer 1

The Correct Option is A

We solve the system of equations. We will use the method of substitution or elimination. Multiply the first equation by 7 and the second by 3 to eliminate one variable. \[ 7(7x + 3y) = 7(12) \implies 49x + 21y = 84 \] \[ 3(3x + 7y) = 3(6) \implies 9x + 21y = 18 \] Subtract the second equation from the first to eliminate \(y\): \[ (49x + 21y) - (9x + 21y) = 84 - 18 \implies 40x = 66 \implies x = \frac{66}{40} = \frac{3}{2} \] Substitute \(x = \frac{3}{2}\) into one of the original equations to find \(y\): \[ 7\left(\frac{3}{2}\right) + 3y = 12 \implies \frac{21}{2} + 3y = 12 \implies 3y = 12 - \frac{21}{2} = \frac{24}{2} - \frac{21}{2} = \frac{3}{2} \implies y = \frac{1}{2} \] Thus, \(x - y = \frac{3}{2} - \frac{1}{2} = \frac{2}{3}\).
Final Answer: \[ \boxed{\frac{2}{3}} \]