Solving Systems of Equations by Graphing Practice

Question

Complete this assessment to review what you've learned. It will not count toward your grade.
1 of 51 of 5 Items
Question
Use the image to answer the 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.

(1 point)
The intersection point is (
).

Answer 1

To find the solution to the system of equations given by the lines \( x - 5y = 3 \) and \( 3x - 2y = -4 \), we will rewrite the equations in slope-intercept form (\( y = mx + b \)) and find their intersection point.

1. For the first equation: \[ x - 5y = 3 \] Rearranging gives: \[ -5y = -x + 3 \] \[ y = \frac{1}{5}x - \frac{3}{5} \]

2. For the second equation: \[ 3x - 2y = -4 \] Rearranging gives: \[ -2y = -3x - 4 \] \[ y = \frac{3}{2}x + 2 \]

Now, we need to find the intersection point by setting the two equations for \( y \) equal to each other: \[ \frac{1}{5}x - \frac{3}{5} = \frac{3}{2}x + 2 \]

To eliminate the fractions, we can multiply through by 10: \[ 10 \left(\frac{1}{5}x\right) - 10 \left(\frac{3}{5}\right) = 10 \left(\frac{3}{2}x\right) + 10(2) \] This simplifies to: \[ 2x - 6 = 15x + 20 \]

Now, combine like terms: \[ 2x - 15x = 20 + 6 \] \[ -13x = 26 \] \[ x = -2 \]

Now, we can substitute \( x = -2 \) back into one of the original equations to find \( y \). We'll use \( y = \frac{3}{2}x + 2 \): \[ y = \frac{3}{2}(-2) + 2 \] \[ y = -3 + 2 \] \[ y = -1 \]

Thus, the intersection point, which is the solution to the system of equations, is \((-2, -1)\).

So, the answer is: The intersection point is \((-2, -1)\).