Ignore $r$ for a moment. Take $10x - 6y = 4$ and solve for $y$ in terms of $x$. This will give you a formula for all points on the line.

Once you have $y$ written as a function of $x$, think of $x$ as the free variable $r$ from the answer choices. Which option has first coordinate equal to that free variable and second coordinate equal to your expression?

Make sure the relationship you find works for every real value of $r$, not just one specific value. If plugging into an equation forces $r$ to be a single number, that choice is not correct.

In Desmos, enter the two equations on separate lines:

• 10x - 6y = 4
• -15x + 9y = -6

You should see that the two graphs overlap exactly, meaning they represent the same line.

Click on each equation in Desmos and use the wrench or settings to have Desmos display the equation solved for $y$. Desmos will rewrite both equations as the same line of the form

Read off the exact values of $m$ and $b$ that Desmos shows.

Think of $x$ as the free variable (which the problem calls $r$). A general point on the line has the form $(x, mx + b)$, so in terms of $r$ it would look like $(r, m r + b)$. Look for the choice where:

• the first coordinate is just $r$, and
• the second coordinate matches the line’s expression $mx + b$ with $x$ replaced by $r$.

In Desmos, define a slider r and then plot each answer choice as a point depending on r (for example, (r, (5r-2)/3) with your actual $m$ and $b$ values). Move the slider and see which option’s point always stays on the line for every value of r. That option is the correct one.

Compare the two equations:

Multiply the first equation by $-1.5$ (or $-\tfrac{3}{2}$):

So the second equation is just $-1.5$ times the first. That means the two equations describe the same line, and any point that satisfies one automatically satisfies the other.

Use the first equation to express $y$ in terms of $x$:

Isolate $y$:

So any point on this line must satisfy

Since $x$ can be any real number, the general point $(x,y)$ on this line can be written as

This describes all points that lie on both graphs, because the graphs are the same line.

In the answer choices, the free variable is called $r$ instead of $x$. So we can rename $x$ as $r$ and write the general solution point as

This matches choice D, so $(r, \, \frac{5r-2}{3})$ is the point that lies on the graph of each equation for every real number $r$.