The lines and intersect at point . If point also lies on the line , what is the value of ?

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SAT Linear Equations in Two Variables Question 105 | Free SAT Prep | Aniko

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Question 105·Hard·Linear Equations in Two Variables

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The lines $y = mx + 4$ and $y = -2x + 1$ intersect at point $P$ . If point $P$ also lies on the line $3y + 2x = 5$ , what is the value of $m$ ?

For SAT problems where a point lies on multiple lines, treat the point’s coordinates as the key: first solve a system using only the equations that do not contain the unknown parameter (here, $m$ ) to find that point, then plug the coordinates into the remaining equation to solve for the parameter. This avoids guessing and keeps the work organized: system first, substitution second, solve last.

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Use the point that lies on all three lines

Point $P$ lies on each of the three lines, so its coordinates must satisfy all three equations at once. Start by finding those coordinates.

Choose which two equations to solve first

Because $m$ only appears in $y=mx+4$ , first use the other two equations, $y=-2x+1$ and $3y+2x=5$ , to find the coordinates of $P$ .

Solve the system without m

Substitute the expression for $y$ from $y=-2x+1$ into $3y+2x=5$ , solve for $x$ , then use that $x$ value to find $y$ .

Use the point you found to get m

After you know $P=(x,y)$ , plug those values into $y=mx+4$ and solve that equation for $m$ .

Desmos Guide

Graph the two lines without m

In Desmos, enter y = -2x + 1 and y = (5 - 2x)/3 (this is $3y+2x=5$ solved for $y$ ). Find and click their intersection point; note its coordinates $(x,y)$ .

Use the intersection coordinates to compute m

Take the intersection coordinates $(x,y)$ you found and, in a new Desmos expression line, type (y - 4)/x. The resulting value shown by Desmos is the value of $m$ that makes the line $y = mx + 4$ pass through that same point.

Step-by-step Explanation

Understand what point P represents

Point $P$ is the intersection of the lines $y=mx+4$ and $y=-2x+1$ , and we are also told that $P$ lies on the line $3y+2x=5$ .

That means the coordinates of $P$ must satisfy all three equations at the same time.

Find the coordinates of P using the two equations that do not contain m

Use the two equations that have only $x$ and $y$ :

$y=-2x+1$
$3y+2x=5$

Substitute $y=-2x+1$ into $3y+2x=5$ :

\begin{aligned} 3(-2x+1)+2x &= 5 \\ -6x+3+2x &= 5 \\ -4x+3 &= 5 \\ -4x &= 2 \\ x &= -\frac{1}{2} \end{aligned}

Now plug $x=-\tfrac{1}{2}$ into $y=-2x+1$ :

y = -2\left(-\frac{1}{2}\right) + 1 = 1 + 1 = 2

So point $P$ is $\left(-\tfrac{1}{2}, 2\right)$ .

Use point P in the equation with m to solve for m

Now use that $P\left(-\tfrac{1}{2}, 2\right)$ lies on the line $y=mx+4$ .

Substitute $x=-\tfrac{1}{2}$ and $y=2$ into $y=mx+4$ :

2 = m\left(-\frac{1}{2}\right) + 4

Solve for $m$ :

\begin{aligned} 2 &= -\frac{m}{2} + 4 \\ 2 - 4 &= -\frac{m}{2} \\ -2 &= -\frac{m}{2} \\ 2 &= \frac{m}{2} \\ m &= 4 \end{aligned}

So the value of $m$ is $4$ .

Strategy

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Desmos Guide

Step-by-step Explanation

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Strategy

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Desmos Guide

Step-by-step Explanation