Question

1. what is the most efficient method to solve this system?

5x - 3y = 11
5x + 2y = 6
a. solve the top equation for x and use substitution
b. solve the bottom equation for y and use substitution
c. add the equations to eliminate x
d. subtract the equations to eliminate x

Explanation:

To determine the most efficient method, we analyze the system of equations \(5x - 3y = 11\) and \(5x + 2y = 6\). Both equations have the same coefficient for \(x\) (which is \(5\)). The elimination method works by eliminating a variable. If we subtract the two equations (subtract the bottom equation from the top or vice versa), the \(x\)-terms will cancel out because \(5x - 5x = 0\). Let's check:

Top equation: \(5x - 3y = 11\)

Bottom equation: \(5x + 2y = 6\)

Subtract the bottom equation from the top equation: \((5x - 3y) - (5x + 2y) = 11 - 6\)

Simplify: \(5x - 3y - 5x - 2y = 5\) → \(-5y = 5\), which eliminates \(x\) and allows us to solve for \(y\) directly.

Option A and B suggest substitution, which would involve more steps (solving for a variable and substituting) compared to elimination here. Option C suggests adding the equations, but adding \(5x - 3y\) and \(5x + 2y\) would give \(10x - y = 17\), which doesn't eliminate \(x\). Option D, subtracting the equations, eliminates \(x\) efficiently.

Answer:

D. Subtract the equations to eliminate x