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.

(1 point)
The intersection point is (
).

Answer 1

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

Starting with the first equation:

1. \( x - 5y = 7 \)

We can express \( x \) in terms of \( y \):

\[ x = 5y + 7 \]

Now substituting \( x \) into the second equation:

2. \( 3x - 2y = -4 \)

Substituting \( x = 5y + 7 \):

\[ 3(5y + 7) - 2y = -4 \]

Expanding this:

\[ 15y + 21 - 2y = -4 \]

Combining like terms:

\[ 13y + 21 = -4 \]

Now, isolating \( y \):

\[ 13y = -4 - 21 \] \[ 13y = -25 \] \[ y = -\frac{25}{13} \]

Now that we have \( y \), we substitute back to find \( x \):

\[ x = 5\left(-\frac{25}{13}\right) + 7 \] \[ x = -\frac{125}{13} + 7 \] \[ x = -\frac{125}{13} + \frac{91}{13} \] \[ x = -\frac{34}{13} \]

Thus, the solution to the system of equations is:

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

As a decimal approximation, this would be approximately:

\[ (-2.615, -1.923) \]

Therefore, the intersection point is:

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

Answer 2

can you make it easier

Answer 3

Sure! Let's simplify the process of finding the intersection point of the two equations, \(x - 5y = 7\) and \(3x - 2y = -4\).

Step 1: Solve the first equation for \(x\)

From the first equation:

\[ x - 5y = 7 \]

We can express \(x\) in terms of \(y\):

\[ x = 5y + 7 \]

Step 2: Substitute into the second equation

Next, we take the second equation, \(3x - 2y = -4\), and substitute \(x\) from the first equation:

\[ 3(5y + 7) - 2y = -4 \]

Step 3: Simplify

Now we will simplify the equation:

\[ 15y + 21 - 2y = -4 \] \[ 13y + 21 = -4 \]

Step 4: Isolate \(y\)

Now isolate \(y\):

\[ 13y = -4 - 21 \] \[ 13y = -25 \] \[ y = -\frac{25}{13} \]

Step 5: Substitute to find \(x\)

Now substitute \(y\) back into the equation for \(x\):

\[ x = 5\left(-\frac{25}{13}\right) + 7 \] \[ x = -\frac{125}{13} + \frac{91}{13} \] \[ x = -\frac{34}{13} \]

Final Answer

So the intersection point of the two lines is:

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

As approximate decimal values, this is about:

\[ (-2.615, -1.923) \]

Conclusion

Thus, the intersection point is:

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

If you just need the coordinates, it can be written as:

\[ (-2.615, -1.923) \]

Let me know if you need anything else or further clarification!