In the -plane, a line passes through the points and . If the slope of is , what is the value of ?

Solution available with step-by-step explanation

SAT Linear Equations in Two Variables Question 39 | Free SAT Prep | Aniko

00:00

Question 39·Hard·Linear Equations in Two Variables

0 / 140

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 $p$ ?

For line-slope problems with variables, always start by writing the slope formula $\frac{y_2 - y_1}{x_2 - x_1}$ using the two given points, being careful with parentheses around expressions that contain variables. Simplify the numerator and denominator, set this fraction equal to the given slope, then clear the fraction and solve the resulting linear equation step by step, watching signs when distributing negatives. If time permits, quickly plug your solution back into the coordinates and recompute the slope to confirm it matches the value in the question.

Hints

Click me!

Use the slope formula

Write the slope of a line through $(p+3, 2p-1)$ and $(6, p)$ using the formula $\frac{y_2 - y_1}{x_2 - x_1}$ . Keep parentheses around any expressions that include $p$ .

Simplify carefully

After you write the slope expression, simplify the numerator and denominator separately. What do you get for $p - (2p - 1)$ and for $6 - (p+3)$ ?

Set equal to the given slope

Once the slope expression is simplified, set it equal to $-2$ and solve the resulting equation for $p$ . Be careful distributing the negative sign when you clear the fraction.

Desmos Guide

Enter the slope expression as a function

In Desmos, treat $p$ as $x$ . Type the expression y = (x - (2x - 1)) / (6 - (x + 3)) to graph the slope of the line in terms of $x$ .

Graph the given slope

On a new line, type y = -2 to graph the horizontal line with slope $-2$ .

Find the value of p

Look for the point where the two graphs intersect and tap it; the $x$ -coordinate of this intersection is the value of $p$ that makes the slope equal to $-2$ .

Step-by-step Explanation

Write the slope in terms of p

The slope of a line through two points $(x_1, y_1)$ and $(x_2, y_2)$ is

\text{slope} = \frac{y_2 - y_1}{x_2 - x_1}.

Here the points are $(p+3,\,2p-1)$ and $(6,\,p)$ , so the slope is

\frac{p - (2p - 1)}{6 - (p+3)}.

We are told this slope equals $-2$ , so we will set this expression equal to $-2$ and solve for $p$ .

Simplify the slope expression

Simplify the numerator:

p - (2p - 1) = p - 2p + 1 = 1 - p.

Simplify the denominator:

6 - (p+3) = 6 - p - 3 = 3 - p.

So the equation becomes

\frac{1 - p}{3 - p} = -2.

Clear the fraction and solve the linear equation

Multiply both sides of

\frac{1 - p}{3 - p} = -2

by $(3 - p)$ to clear the denominator:

1 - p = -2(3 - p).

Distribute on the right side:

1 - p = -6 + 2p.

Now collect like terms by adding $p$ to both sides and adding $6$ to both sides:

\begin{aligned} 1 + 6 &= -6 + 2p + p \\ 7 &= 3p. \end{aligned}

Solve for p and state the answer

From

7 = 3p,

divide both sides by $3$ :

p = \frac{7}{3}.

So the value of $p$ is $\dfrac{7}{3}$ , which corresponds to choice C.

Strategy

Hints

Desmos Guide

Step-by-step Explanation

Study Hints

Strategy

Hints

Desmos Guide

Step-by-step Explanation