Problem 88 - College Algebra Essentials

Question

Explain how to solve a system of equations using the addition method. Use \(3 x+5 y=-2\) and \(2 x+3 y=0\) to illustrate your explanation.

Step-by-Step Solution

Answer

The solution to the system of equations is \(x = 6\) and \(y = -4\).

Step 1: Multiply the equations to make coefficients of y's (or x's) the same

Multiply the first equation by 3 and the second equation by 5. This will result in the new system of equations: \(9x + 15y = -6\) and \(10x + 15y = 0\)

Step 2: Subtract the equations

Subtract the second equation from the first equation to eliminate y: \( (9x - 10x) + (15y - 15y) = -6 - 0\). This results in \(-x = -6.\)

Step 3: Solve for x

To isolate x, multiply both sides of the equation by -1 resulting in \(x = 6\).

Step 4: Substitute x into one of the original equations and solve for y

Substitute \(x = 6\) into the second original equation: \(2*6 + 3y = 0\), this simplifies to \(12 + 3y = 0\). After rearranging, we get \(3y = -12.\) To solve for y, divide both sides by 3 to obtain \(y = -4\).