Once you have the slope of the line through the two points, multiply that slope by 2 to get the slope of line $\ell$.

Write $y = mx + b$ for line $\ell$ using the slope you found. Then plug in $x = 4$ and $y = -3$ (from the point $(4,-3)$) and solve the resulting equation for $b$, which is the y-intercept.

In Desmos, type the expression m1 = (-2 - 2)/(5 - 1) to calculate the slope of the line through $(1,2)$ and $(5,-2)$. Note the value of m1 that Desmos shows.

In a new line, type m = 2*m1 to get the slope of line $\ell$. Desmos will display the value of m.

Use the formula $b = y - mx$ with the point $(4,-3)$. In Desmos, type b = -3 - m*4. The value Desmos shows for b is the y-intercept of line $\ell$.

Use the slope formula for the line through $(1,2)$ and $(5,-2)$:

So the slope of the line through the two given points is $-1$.

Line $\ell$ has a slope that is twice the slope you just found.

So if the original slope is $-1$, then the slope of line $\ell$ is

Thus, the slope of line $\ell$ is $-2$.

Use the slope-intercept form $y = mx + b$ for line $\ell$.

You know:

• The slope is $m = -2$.
• The line passes through $(4,-3)$, so $x = 4$ and $y = -3$ satisfy the equation.

Substitute these into $y = mx + b$ to form an equation for $b$:

Now solve this equation for $b$.

Solve the equation

First compute $-2(4)$:

So the equation becomes

Add 8 to both sides:

The y-intercept of line $\ell$ is the value of $b$, which is 5.