Use the image to answer the question.

Question

A coordinate plane with four quadrants shows the x- and y-axes ranging from negative 5 to 5 in increments of 1. A solid line and a dotted line intersect each other. The equation of the solid line is x minus 5 y equals 3. The equation of the dotted line is 3 x minus 2 y equals negative 4. The intersection of both lines is shown at negative 2 on the x-axis and negative 1 on the y-axis in quadrant 3.

Review the graphs of a system of two linear equations in two variables: x−5y=7
and 3x−2y=−4
. Find the solution to both equations.

Answer 1

To find the solution to the system of equations given by \( x - 5y = 7 \) and \( 3x - 2y = -4 \), we need to solve them simultaneously.

Step 1: Solve \( x - 5y = 7 \) for \( x \)

Rearranging the first equation: \[ x = 5y + 7 \]

Step 2: Substitute \( x \) into the second equation

Now, substitute \( x \) in the second equation: \[ 3(5y + 7) - 2y = -4 \] Expanding this gives: \[ 15y + 21 - 2y = -4 \] Simplifying: \[ 13y + 21 = -4 \] Subtracting 21 from both sides: \[ 13y = -4 - 21 \] \[ 13y = -25 \] Dividing by 13: \[ y = -\frac{25}{13} \]

Step 3: Substitute \( y \) back into the equation for \( x \)

Now, substitute \( y \) back into \( x = 5y + 7 \): \[ x = 5\left(-\frac{25}{13}\right) + 7 \] Calculating \( 5\left(-\frac{25}{13}\right) \): \[ x = -\frac{125}{13} + 7 \] Convert 7 into a fraction to combine: \[ 7 = \frac{91}{13} \] Thus: \[ x = -\frac{125}{13} + \frac{91}{13} \] \[ x = \frac{-125 + 91}{13} \] \[ x = \frac{-34}{13} \]

Step 4: Write the solution

The solution to the system of equations is: \[ \left( \frac{-34}{13}, -\frac{25}{13} \right) \]

This point represents the intersection of the two lines described by the equations \( x - 5y = 7 \) and \( 3x - 2y = -4 \).