Question 643·Hard·Algebra

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In the $xy$ -plane, a line $\ell$ passes through the points $\bigl(p+3,\;2p-1\bigr)$ and $(6,\,p)$ . If the slope of $\ell$ is $-2$ , what is the value of $p$ ?

If two points contain the same unknown, their slope gives you an equation for that unknown. Write the change in $y$ over the change in $x$ , keeping the point order the same in both differences. A one-unknown regression in Desmos can then solve the equation. Reversing only one difference changes the slope’s sign.

Hints

Hint 1
The slope is the change in $y$ divided by the change in $x$ . Since the coordinates contain $p$ , keep them as expressions rather than guessing a value. What are the two changes between the points?
Hint 2
Start at the point with $x=p+3$ and end at $(6,p)$ . Subtract in the same order for both coordinates: $p-(2p-1)$ goes on top, and $6-(p+3)$ goes on the bottom. What must that fraction equal?
Hint 3
Desmos treats a plain letter such as $p$ as a slider. To solve for it, replace every $p$ with $x_1$ and use a regression, marked by $\sim$ instead of $=$ . Read the result under PARAMETERS.

Step-by-step

Write the slope equation and solve in Desmos
Step 1
Turn the slope into an equation
A slope is the change in $y$ divided by the change in $x$ . Going from $(p+3,2p-1)$ to $(6,p)$ , the $y$ -change is $p-(2p-1)$ , and the $x$ -change is $6-(p+3)$ . Keep the same point order in both differences; reversing only one would change the sign. The slope is $-2$ , so:
$\frac{p-(2p-1)}{6-(p+3)}=-2.$
Step 2
Solve for the unknown and read its exact value
In Desmos, a plain $p$ would make a slider, so replace every $p$ with $x_1$ . Type $\frac{x_1-(2x_1-1)}{6-(x_1+3)}\sim-2$ ; $\sim$ tells Desmos to find the value that makes the sides equal. Under PARAMETERS, Desmos shows $x_1\approx2.333333$ . Type $x_1$ on the next line and use the fraction button to see $\frac{7}{3}$ . Since $x_1$ stands for $p$ , the value of $p$ is $\frac{7}{3}$ . Choice C.

Question 643·Hard·Algebra

Strategy

Hints

Step-by-step

Lessons that teach this

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Strategy

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Step-by-step

Lessons that teach this