Once you have the slope, plug it into point-slope form, $y - y_1 = m(x - x_1)$, using either of the two given points.

At the $x$-intercept, the $y$-coordinate is $0$. Substitute $y=0$ into your line equation and solve for the $x$-value, which is $k$.

Plot the two given points by typing (-5,8) and (1,3) into Desmos (each on its own line). Then enter the line equation y - 3 = ((3-8)/(1-(-5)))(x - 1) so Desmos draws the line through those points.

On the graph, locate where the line crosses the $x$-axis (where $y=0$). Click that intersection point and read off the $x$-coordinate; that value is $k$.

Use the slope formula with the two points $(-5,8)$ and $(1,3)$:

So the slope of the line is $-\frac{5}{6}$.

Use point-slope form with the point $(1,3)$ and slope $m=-\frac{5}{6}$:

This is an equation for the line that passes through $(-5,8)$ and $(1,3)$.

The point where the line crosses the $x$-axis has coordinates $(k,0)$, so plug $y=0$ into the equation:

Simplify:

Multiply both sides by $6$:

Divide by $-1$:

Distribute the $5$:

Add $5$ to both sides:

Finally, divide both sides by $5$ to solve for $k$:

So the value of $k$ is $\frac{23}{5}$.