In the -plane, line is defined by . A second line is defined by , where and are constants. The two lines intersect at a point that lies on the line . If , what is the value of ?

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SAT Linear Equations in Two Variables Question 120 | Free SAT Prep | Aniko

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Question 120·Hard·Linear Equations in Two Variables

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In the $xy$ -plane, line $\ell$ is defined by $3x - 2y = 7$ . A second line $m$ is defined by $kx + 6y = p$ , where $k$ and $p$ are constants. The two lines intersect at a point that lies on the line $y = 2x - 1$ . If $k = 4$ , what is the value of $p$ ?

When a problem says that an intersection point lies on another line, treat that point as a solution to all of the involved equations. First, pick the two equations that are easiest to solve as a system (here, $3x - 2y = 7$ and $y = 2x - 1$ ) to find the actual coordinates of the intersection. Then plug those coordinates into the remaining line’s equation to solve for any unknown parameter, being very careful with negative signs and basic arithmetic to avoid simple calculation errors.

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Think about what it means to be the same intersection point

The point where lines $\ell$ and $m$ intersect is also on the line $y = 2x - 1$ . What does that tell you about the coordinates $(x, y)$ of this point and the equations it must satisfy?

Find the coordinates of the intersection point first

Use the equations $3x - 2y = 7$ and $y = 2x - 1$ together as a system. Substitute $y$ from the second equation into the first to solve for $x$ , then find $y$ .

Use the point to determine p

Once you know the intersection point $(x, y)$ , remember that it also lies on line $m$ , whose equation is $kx + 6y = p$ with $k = 4$ . Substitute your $x$ and $y$ into this equation to solve for $p$ .

Desmos Guide

Graph the two lines that determine the intersection point

Enter the equations 3x - 2y = 7 and y = 2x - 1 into Desmos. Click on their point of intersection; Desmos will display its exact coordinates $(x_1, y_1)$ .

Use the intersection point to compute p

Line $m$ with $k = 4$ has equation $4x + 6y = p$ . In a new Desmos expression line, type 4*(x1) + 6*(y1) using the numerical $x_1$ and $y_1$ values from the intersection point. The output of this expression is the value of $p$ .

Step-by-step Explanation

Use the fact that the intersection point lies on all three lines

If two lines intersect at a point that lies on a third line, then that single point $(x, y)$ satisfies all three equations.

So the intersection point of lines $\ell$ and $m$ must satisfy:

$3x - 2y = 7$ (line $\ell$ )
$y = 2x - 1$ (given line)

First, use these two equations to find the coordinates of the intersection point.

Find the intersection of $3x - 2y = 7$ and $y = 2x - 1$

Substitute $y = 2x - 1$ into $3x - 2y = 7$ :

\begin{aligned} 3x - 2(2x - 1) &= 7 \\ 3x - 4x + 2 &= 7 \\ -x + 2 &= 7 \\ -x &= 5 \\ x &= -5 \end{aligned}

Now plug $x = -5$ into $y = 2x - 1$ :

y = 2(-5) - 1 = -10 - 1 = -11

So the intersection point of the first line and $y = 2x - 1$ is $(-5, -11)$ . This is also the intersection point of lines $\ell$ and $m$ .

Write line m with k = 4 and solve for p

Line $m$ is given by $kx + 6y = p$ . With $k = 4$ , this becomes

4x + 6y = p

Because $(-5, -11)$ lies on line $m$ , substitute $x = -5$ and $y = -11$ into this equation:

\begin{aligned} p &= 4(-5) + 6(-11) \\ &= -20 - 66 \\ &= -86 \end{aligned}

So the value of $p$ is $-86$ , which corresponds to choice B.

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Step-by-step Explanation

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Desmos Guide

Step-by-step Explanation