Problem 3 - Algebra 1: Concepts and Skills

Question

Choose a method to solve the linear system. Explain your choice. $$ \begin{aligned} &3 x+5 y=25\\\ &2 x-6 y=12 \end{aligned} $$

Step-by-Step Solution

Answer

The solution to the system of equations is \(x = 9.17\) and \(y = -0.5\).

Step 1: Set Up the Equations

To use the elimination method, both equations need to be written in a form where like terms are lined up vertically. The equations are already in this form:\n \[ \begin{aligned} &3x + 5y = 25\ &2x - 6y = 12 \end{aligned}\]

Step 2: Multiply Equations to Make Coefficients Opposites

In order to eliminate one of the variables, the coefficients in front of \(x\) or \(y\) in both equations need to be opposites. Multiplying the first equation by 2 and the second equation by 3, the coefficients of \(x\) will become opposites:\n \[ \begin{aligned} &6x + 10y = 50\ &6x - 18y = 36 \end{aligned}\]

Step 3: Subtract Equations to Eliminate One Variable

Subtracting the second equation from the first will eliminate \(x\), allowing to solve for \(y\): \[ (-6x + 18y) - (6x - 10y) = 36 - 50 \] Simplifying this gives: \[ 28y = -14 \]

Step 4: Solve for the Remaining Variable

Now that there is only one variable in the equation, \(y\) can be solved for by dividing both sides of the equation by 28, giving \(y = -0.5\)

Step 5: Substitute \(y\) value into Original Equation to Solve for \(x\)

Substitute \(y = -0.5\) into the first original equation: \[ 3x + 5(-0.5) = 25 \] Simplifying this gives: \[ 3x = 25 + 2.5 = 27.5 \] And finally dividing through by 3 gives \(x = 9.17\)