Rewrite $4x - 5y = 20$ in the form $y = mx + b$ to identify the slope of line k. How is the slope of a line perpendicular to this related to $m$?

Once you know the slope of line m and the coordinates of point P, plug them into either point-slope form $y - y_1 = m(x - x_1)$ or into $y = mx + b$ to get an equation for line m.

After finding an equation for line m, rearrange it into the form $Ax + By = C$ with integer coefficients and compare it with the answer options.

Type 4x - 5y = 20 and y = -2x + 3 into Desmos. Tap on their intersection to see the coordinates of point $P$; this is the point through which line m must pass.

Either solve $4x - 5y = 20$ for $y$ (so you can see the slope $4/5$), or visually note that slope from the graph. Then take the negative reciprocal to find the slope that a perpendicular line must have; this is the slope that line m needs.

Enter each option (for example, 10x + 8y = 9) into Desmos one at a time. For each, solve it for $y$ within Desmos or visually inspect the graph to check two things: (1) does the line’s slope match the perpendicular slope you found, and (2) does the line pass exactly through point $P$ from step 1?

The correct choice is the one whose graph both has the perpendicular slope from step 2 and goes through point $P$ from step 1; read off which equation that is from the answer options.

Line k: $4x - 5y = 20$

Line j: $y = -2x + 3$

Substitute $y = -2x + 3$ into the equation for line k:

Simplify:

Now substitute $x = 5/2$ into $y = -2x + 3$:

So the lines intersect at $P = (5/2, -2)$. This point lies on line m as well.

Start with line k:

Solve for $y$ to get slope-intercept form.

Subtract $4x$ from both sides:

Divide by $-5$:

The slope of line k is $4/5$.

For a perpendicular line, the slope is the negative reciprocal: flip the fraction and change the sign.

So the slope of line m is $-5/4$. Line m must have slope $-5/4$ and pass through $P = (5/2, -2)$.

Use slope-intercept form $y = mx + b$ with $m = -5/4$.

Plug in the coordinates of point $P = (5/2, -2)$ to solve for $b$:

Compute the product:

So:

Write $-2$ as $-16/8$:

Add $25/8$ to both sides:

Thus line m in slope-intercept form is:

The answer choices are in standard form $Ax + By = C$, with integer coefficients.

Start from the slope-intercept form of line m:

Clear the denominators by multiplying every term by 8:

Now move $-10x$ to the left side by adding $10x$ to both sides:

This matches choice A), so the correct equation of line m is $10x + 8y = 9$.