You should now have an expression for $g(3t - 5)$ in terms of $t$, $k$, and $m$. Set this equal to $2t + 7$. Since these are equal for every real $t$, how must the coefficients of $t$ and the constant terms compare?

From comparing coefficients you should get two linear equations in $k$ and $m$. Solve the simpler one (in just $k$) first, then substitute that value of $k$ into the other equation to find $m$.

Once you know $k$ and $m$, plug them into $k - m$, simplify carefully with fractions, and then multiply by 3.

After you have found $k = 2/3$ and $m = 31/3$ by hand, type the expression 3*((2/3)-(31/3)) into Desmos. The value that Desmos outputs is the value of $3(k - m)$ for this problem.

We know $g(t) = kt + m$. So, to find $g(3t - 5)$, replace $t$ by $3t - 5$ in the formula for $g$:

The problem says $g(3t - 5) = 2t + 7$ for all real $t$.

From Step 1, we have:

Because these are equal for all $t$, the coefficients of $t$ must be equal and the constant terms must be equal:

• Coefficient of $t$: $3k = 2$.
• Constant term: $-5k + m = 7$.

First solve $3k = 2$:

Now plug this value of $k$ into $-5k + m = 7$:

Add $\frac{10}{3}$ to both sides:

So $k = \frac{2}{3}$ and $m = \frac{31}{3}$.

Now find $3(k - m)$ using the values of $k$ and $m$:

Inside the parentheses:

Now multiply by 3:

So the value of $3(k - m)$ is $-29$, which corresponds to choice B.