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

no

Answer 2

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

**Step 1: Solve for \( y \) in terms of \( x \) using the first equation.**

Starting with the equation:

\[
x - 5y = 7
\]

Rearranging to isolate \( y \):

\[
-5y = 7 - x
\]

\[
y = \frac{x - 7}{5}
\]

**Step 2: Substitute \( y \) in the second equation.**

Now we will substitute this expression for \( y \) into the second equation \( 3x - 2y = -4 \):

\[
3x - 2\left(\frac{x - 7}{5}\right) = -4
\]

**Step 3: Clear the fraction by multiplying the entire equation by 5.**

\[
5(3x) - 2(x - 7) = -4 \cdot 5
\]

\[
15x - 2(x - 7) = -20
\]

Expanding the equation:

\[
15x - 2x + 14 = -20
\]

Combining like terms:

\[
13x + 14 = -20
\]

**Step 4: Solve for \( x \).**

Subtracting 14 from both sides gives:

\[
13x = -20 - 14
\]

\[
13x = -34
\]

Dividing by 13:

\[
x = -\frac{34}{13}
\]

**Step 5: Substitute \( x \) back to find \( y \).**

Now substitute \( x = -\frac{34}{13} \) back into the equation for \( y \):

\[
y = \frac{-\frac{34}{13} - 7}{5}
\]

Converting 7 to a fraction with a common denominator of 13:

\[
y = \frac{-\frac{34}{13} - \frac{91}{13}}{5}
\]

\[
y = \frac{-\frac{125}{13}}{5}
\]

\[
y = -\frac{125}{65} = -\frac{25}{13}
\]

**Step 6: Write the solution as an ordered pair.**

Thus, the intersection point (solution to the system of equations) is:

\[
\left(-\frac{34}{13}, -\frac{25}{13}\right)
\]

Final answer: The intersection point is \(\left(-\frac{34}{13}, -\frac{25}{13}\right)\).