the line y=mx+n is the reflection of 5x-3y = 10 about the line y=x

To find the equation of a line that is the reflection of \(5x - 3y = 10\) about the line \(y = x\), follow these steps:

### 1. **Rewrite the given line in slope-intercept form**:
First, rewrite the given line \(5x - 3y = 10\) in the slope-intercept form \(y = mx + b\).

\[
5x - 3y = 10
\]
\[
-3y = -5x + 10
\]
\[
y = \frac{5}{3}x - \frac{10}{3}
\]

So, the equation of the line is:
\[
y = \frac{5}{3}x - \frac{10}{3}
\]

### 2. **Reflect the line over \(y = x\)**:
When a line \(y = mx + n\) is reflected over the line \(y = x\), the slope of the reflected line becomes the reciprocal of the original slope \(m\), and the equation transforms to \(x = my + n'\).

To reflect \(y = \frac{5}{3}x - \frac{10}{3}\), we switch \(x\) and \(y\):
\[
x = \frac{5}{3}y - \frac{10}{3}
\]

### 3. **Solve for \(y\) to obtain the equation in standard form**:
Now, solve for \(y\) to get the equation of the reflected line:
\[
\frac{5}{3}y = x + \frac{10}{3}
\]
\[
y = \frac{3}{5}x + 2
\]

### 4. **Final equation**:
Thus, the equation of the reflected line is:
\[
y = \frac{3}{5}x + 2
\]

This is the equation of the line after reflecting \(5x - 3y = 10\) about the line \(y = x\).

Would you like more details or have any questions about this solution?

### Relative Questions:
1. What happens when a line is reflected over the line \(y = -x\)?
2. How would you reflect a line over the y-axis?
3. What is the geometric significance of reflecting a line over another line?
4. How do you find the intersection point of two lines algebraically?
5. How can you verify that the reflection is correct by substituting a point?

### Tip:
When reflecting a line over \(y = x\), simply swap \(x\) and \(y\) in the equation and solve for the new \(y\) to get the reflected line equation.

Learn how to reflect the line 5x - 3y = 10 over the line y = x with a detailed step-by-step solution. This problem covers the concepts of line reflection, slope-intercept form, and coordinate geometry. Suitable for high school students.

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