Problem 71 - 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 of the system of equations is \(x = -6/19\) and \(y = -22/95\).

Step 1: Multiply Equations

Firstly, multiply the two equations by necessary multiples such that the coefficients of y's in both equations will cancel out each other when they are added together. In this case, multiply the first equation by 3 and the second by 5. After multiplication, the two equations become \(9x + 15y = -6\) and \(10x + 15y = 0\).

Step 2: Add the Equations

With the first equation \(9x + 15y = -6\) and the second equation \(10x + 15y = 0\), add the two equations together. The term \(15y\) in both equations will cancel out each other, resulting in \(19x = -6\).

Step 3: Solve for One Variable

To solve for x, divide both sides of the equation by 19, so \(x = -6/19\).

Step 4: Substitute Back and Solve for the Other Variable

Now, with the value of x as -6/19, substitute x in the first original equation \(3x +5y = -2\) to find the value of y. This results in the equation \(5y = -2 - 3(-6/19) = -2 + 18/19 = -22/19\). Then solve for y, \(y = -22/95\).