Problem 85 - Introductory Algebra for College Students

Question

The point of intersection of the graphs of the equations \(A x-3 y=16\) and \(3 x+B y=7\) is \((5,-2) .\) Find \(A\) and \(B\)

Step-by-Step Solution

Answer

The values of A and B are 2 and 4 respectively.

Step 1: Substituting Point of Intersection into the First Equation

Substitute \(x = 5\) and \(y = -2\) into the first equation. We get: \(A(5) - 3(-2) = 16\) which simplifies to \(5A + 6 = 16\). Therefore, \(A = \frac{16 - 6}{5} = 2.\)

Step 2: Substituting Point of Intersection into the Second Equation

Substitute \(x = 5\) and \(y = -2\) into the second equation. We get: \(3(5) + B(-2) = 7\) which simplifies to \(15 - 2B = 7\). Therefore, \(B = \frac{15 - 7}{2} = 4.\)

Step 3: Summary of Solution

By substituting the point of intersection into each equation, we have found that \(A = 2\) and \(B = 4\).