Question 177·Hard·Math

Distribute $t$ on the left side to rewrite $t(x - 1) + 3$ and then group the $x$ terms and constant terms.

Once both sides are written as something times $x$ plus a constant, ask: what conditions on $t$ and $d$ make these two linear expressions exactly the same for all $x$?

After you find the specific values of $t$ and $d$ that make the equation an identity, see which roman-numeral statements match those values.

In Desmos, enter the expression y = t*(x - 1) + 3 - (2*x + d) and create sliders for $t$ and $d$. This graph shows the difference between the two sides of the equation.

Adjust the sliders for $t$ and $d$ and watch the graph. The equation has infinitely many solutions exactly when the graph of the difference is the horizontal line $y = 0$ for all $x$ (meaning the two sides are equal for every $x$). Identify which specific slider values make this happen.

Once you find the values of $t$ and $d$ that make the graph coincide with $y = 0$, compare those values to statements I, II, and III to see which statements match what you observed in Desmos.

The given equation is

and $x$ is the variable. For a linear equation in $x$ to have infinitely many solutions, it must be true for every value of $x$. That only happens if the expressions on both sides are identical as functions of $x$ (same $x$-coefficient and same constant term).

First expand the left side:

So the equation becomes

Now you can clearly see the $x$-coefficient and constant term on each side.

For the equation

to be true for all $x$:

• The coefficients of $x$ must be equal: $t = 2$.
• The constant terms must be equal: $3 - t = d$.

Substitute $t = 2$ into $3 - t = d$:

So the only way the equation has infinitely many solutions is when $t = 2$ and $d = 1$.

We found that $t = 2$ and $d = 1$ must both be true for infinitely many solutions.

• Statement I: $t = 2$ → true.
• Statement II: $d = 1$ → true.
• Statement III: $d = 5$ → false, because $d$ must equal $1$.

Therefore, the correct choice is I and II only.