Question 36·Hard·Systems of Two Linear Equations in Two Variables

The phrase "for each real number $t$" means the correct choice must describe all solutions to the system as $t$ varies. Think of $t$ as a placeholder for one of the coordinates, like $y$ or $x$.

Try setting $y = t$ and substitute into $4x - 6y = 9$. Solve this equation for $x$ in terms of $t$, then look for the answer choice that uses those same expressions for $x$ and $y$.

Pick one answer choice and plug its $x$ and $y$ expressions into $4x - 6y = 9$. Does the equation simplify to a true statement (like $9 = 9$) for every $t$, or does it depend on $t$?

In Desmos, type 4x - 6y = 9. You do not need to enter the second equation because it describes the same line (it is just $-2$ times the first equation).

In an empty expression line, type t = 0 and Desmos will offer to create a slider. Accept the slider so you can vary $t$.

For each option, enter a point that depends on $t$, for example:

• A: ( (3t+9)/4 , t )
• B: ( (9+6t)/4 , t )
• C: ( t , (4t+9)/6 )
• D: ( (9-6t)/4 , t ) Desmos will plot each of these as a point that moves when you change $t$ with the slider.

Move the $t$ slider through several values (both positive and negative). For each answer choice, watch whether its corresponding point always remains on the line 4x - 6y = 9. The correct choice is the one whose point stays on the line for every value of $t$ you try.

Look at the coefficients in the system:

Multiply the first equation by $-2$:

This is exactly the second equation. That means both equations represent the same line, so any point that lies on one equation automatically lies on the other.

Because there are infinitely many points on a line, the answer choices use a parameter $t$ to describe them.

We can choose one variable to equal $t$ and then solve for the other variable. The answer choices mostly use $y = t$, so we will do the same: let

Substitute $y = t$ into the first equation $4x - 6y = 9$:

Now solve this equation for $x$:

From

divide both sides by $4$:

So every point on the line (and therefore on both equations) can be written as

This exactly matches choice B) $\bigl(\tfrac{9+6t}{4}\,,\,t\bigr)$, so B is the correct answer.