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.