Question 192·Hard·Algebra

Use the two points you found and apply the slope formula $m=\dfrac{y_2-y_1}{x_2-x_1}$. Be consistent: use the same order of points in the numerator and denominator.

After plugging into the slope formula, simplify the numerator and denominator separately. Watch your minus signs when subtracting expressions like $(-2p+11)-(4p+5)$ and $(5p+1)-(3p-2)$.

Once you simplify, see if the numerator can be factored to match the forms in the answer choices, which all involve $(p-1)$, $(p+1)$, or similar expressions.

In Desmos, type p = 2 (or another simple number that does not make any denominator zero) to assign a specific value to $p$ so you can work numerically.

Enter m = ((-2*p + 11) - (4*p + 5)) / ((5*p + 1) - (3*p - 2)). Desmos will display a numeric value for m, which is the slope of the line for your chosen $p$.

In Desmos, type expressions for each option using the same $p$ value, for example: A = 6*(p-1)/(2*p+3), B = 6*(p+1)/(2*p-3), C = (p-1)/(2*p+3), and D = -6*(p-1)/(2*p+3). Compare each expression’s value to the slope m from step 2.

The correct answer choice is the one whose value exactly matches the slope m you computed from the two points for that same value of $p$.

For a function $f$, the statement $f(a)=b$ means the graph of $f$ passes through the point $(a,b)$.

So from the problem:

• $f(3p-2)=4p+5$ gives the point $(3p-2,\,4p+5)$.
• $f(5p+1)=-2p+11$ gives the point $(5p+1,\,-2p+11)$.

These are two points on the line representing the linear function $f$.

Let $(x_1,y_1)=(3p-2,\,4p+5)$ and $(x_2,y_2)=(5p+1,\,-2p+11)$.

The slope $m$ of a line through two points is:

Substitute the coordinates:

Now simplify the numerator and denominator separately.

First, simplify the numerator:

Next, simplify the denominator:

You now have the simplified numerator and denominator needed to write the slope.

Using the simplified parts from the previous step, the slope of $f$ in terms of $p$ is

Comparing with the options, this matches choice D) $\dfrac{-6(p-1)}{2p+3}$.