Find the intersection of the solution sets of the following linear equations: \(2x - 3y = 6\) and \(5x + 4y = 20\).
Answer
Substitute \(x = 2\) into each of the original equations to find the corresponding y-values. For \(2x - 3y = 6\), this gives \(y = 0\), and for \(5x + 4y = 20\), this gives \(y = 0\) as well.
Step 1 ：First, solve each equation for y: \(y = \frac{2x - 6}{3}\) and \(y = \frac{20 - 5x}{4}\).
Step 2 ：Then, set the two expressions for y equal to each other to find the x-value of the intersection: \(\frac{2x - 6}{3} = \frac{20 - 5x}{4}\). Solving this equation gives \(x = 2\).
Step 3 ：Substitute \(x = 2\) into each of the original equations to find the corresponding y-values. For \(2x - 3y = 6\), this gives \(y = 0\), and for \(5x + 4y = 20\), this gives \(y = 0\) as well.
