Plug $(k, 5k+3)$ and $(2k, 7k-1)$ into the slope formula and write the slope as a fraction in terms of $k$.

Be careful with parentheses and minus signs when simplifying $(7k-1) - (5k+3)$, and also simplify the denominator $2k - k$.

Once the slope is written in terms of $k$, set that expression equal to 4 and solve the resulting equation for $k$, remembering that $k \neq 0$.

In Desmos, type (2k - 4)/k as an expression. Desmos will treat k as a variable, and you can add a slider for k if it does not appear automatically.

In a new line, type y = 4 to represent the constant slope value, and in another line type y = (2k - 4)/k but then replace k with x so Desmos can graph it as y = (2x - 4)/x. You are now graphing the slope expression as a function of $x$.

Look for the intersection point of the graph of y = (2x - 4)/x with the horizontal line y = 4. The x-coordinate of this intersection is the value of $k$ that makes the slope 4.

The slope $m$ of a line that passes through points $(x_1,y_1)$ and $(x_2,y_2)$ is

Here, the two points are $(k, 5k+3)$ and $(2k, 7k-1)$, and the slope is given as 4.

Take $(x_1, y_1) = (k, 5k+3)$ and $(x_2, y_2) = (2k, 7k-1)$.

Then the slope in terms of $k$ is

First simplify the numerator:

• $(7k - 1) - (5k + 3) = 7k - 1 - 5k - 3 = 2k - 4$.

The denominator is

• $2k - k = k$.

So the slope becomes

The problem says the slope is 4, so

Multiply both sides by $k$ (and remember $k \neq 0$):

Subtract $2k$ from both sides:

Divide both sides by 2:

So, the value of $k$ is $-2$.