Problem 88 - College Algebra

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 \(3x + 5y = -2\) and \(2x + 3y = 0\) is \(x = 6\), \(y = -4\).

Step 1: Multiply to match coefficients

To cancel out a variable, the coefficients of this variable in both equations need to be equal with opposite signs. Mathematicians usually aim to eliminate the variable with the smallest coefficient for simplicity reasons. In this case, it's x. So, first, multiply the first equation by 2 and the second by 3. This produces the following system: \( 6x + 10y = -4 \) and \(6x + 9y = 0\).

Step 2: Subtract to eliminate

Now subtract the second equation from the first to cancel the x variable: \((6x + 10y) - (6x + 9y) = -4 - 0\), which simplifies to \(y = -4\). This represents the value of variable y.

Step 3: Substitute and solve

Now, substitute y = -4 into the original second equation: \(2x + 3(-4) = 0\). This simplifies to \(2x = 12\), and further simplifies to \(x = 6\). This represent the value of variable x.