In the coordinate plane shown, line passes through points and . Point is also shown. Line has equation . Let be the point where lines and intersect. Which choice is the slope of line ?

Solution available with step-by-step explanation

SAT Linear Equations in Two Variables Question 60 | Free SAT Prep | Aniko

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Question 60·Hard·Linear Equations in Two Variables

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In the coordinate plane shown, line $s$ passes through points $A$ and $B$ . Point $T$ is also shown.

Line $r$ has equation $2x+y=11$ . Let $P$ be the point where lines $r$ and $s$ intersect.

Which choice is the slope of line $\overline{PT}$ ?

Treat the graph as information for building a linear equation: read two clear points on the graphed line and write its equation. Then solve the system formed by that equation and the given equation to find the intersection point. Once you have two points, slope is always $\frac{\Delta y}{\Delta x}$ , so compute it carefully using the same point order in both differences.

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Start with line $s$

Use the two labeled points on line $s$ to find its slope.

Write a system

Write line $r$ in the form $y=mx+b$ , and also write line $s$ as $y=mx+b$ .

Find point $P$

Set the two expressions for $y$ equal to find the intersection point $P$ .

Finish with a slope

Use the slope formula between $P$ and point $T$ .

Desmos Guide

Enter line $r$

Graph line $r$ by typing $y=11-2x$ .

Enter line $s$ from the graph

From the graph, use points $A(2,5)$ and $B(6,2)$ to get slope $-\frac{3}{4}$ , then type $y=-\frac{3}{4}x+\frac{13}{2}$ .

Create the intersection point $P$

Click the intersection of the two lines to create point $P$ . Desmos will display its coordinates.

Compute the slope from $P$ to $T$

Define $T=(1,-2)$ . Then type an expression like

\frac{P.y-T.y}{P.x-T.x}

Match the computed value to the answer choices.

Step-by-step Explanation

Read two points on line $s$

From the graph, the labeled points on line $s$ are $A(2,5)$ and $B(6,2)$ . Also, $T(1,-2)$ .

Write an equation for line $s$

Compute the slope of $s$ using $A$ and $B$ :

m_s=\frac{2-5}{6-2}=\frac{-3}{4}

Use point-slope form with point $A(2,5)$ :

\begin{aligned} y-5&=-\frac{3}{4}(x-2)\\ y&=-\frac{3}{4}x+\frac{13}{2} \end{aligned}

Find the intersection point $P$ of lines $r$ and $s$

Rewrite line $r$ :

2x+y=11 \quad \Rightarrow \quad y=11-2x

Set the two expressions for $y$ equal:

11-2x=-\frac{3}{4}x+\frac{13}{2}

Multiply by 4 to clear fractions:

44-8x=-3x+26

So $18=5x$ , giving $x=\frac{18}{5}$ . Then

y=11-2\left(\frac{18}{5}\right)=\frac{19}{5}

Thus, $P\left(\frac{18}{5},\frac{19}{5}\right)$ .

Compute the slope of $\overline{PT}$

Use $P\left(\frac{18}{5},\frac{19}{5}\right)$ and $T(1,-2)$ :

\begin{aligned} m_{PT} &=\frac{\frac{19}{5}-(-2)}{\frac{18}{5}-1}\\ &=\frac{\frac{19}{5}+\frac{10}{5}}{\frac{18}{5}-\frac{5}{5}}\\ &=\frac{\frac{29}{5}}{\frac{13}{5}}\\ &=\frac{29}{13} \end{aligned}

Therefore, the slope of $\overline{PT}$ is $\frac{29}{13}$ .

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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