Question 298·Hard·Algebra

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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 $\overline{PT}$ ?

A line given by two marked points can be paired with another line to find their intersection, the point on both lines. Read the marked coordinates, write an equation for the graphed line, then graph both equations in Desmos and click where they cross. Use that crossing point with the other endpoint to find the requested slope. The rise alone is not a slope; divide it by the horizontal change.

Hints

Hint 1
The marked points on $s$ let you find its slope, the change in $y$ divided by the change in $x$ . Go from $A$ to $B$ in the same order on top and bottom. What slope do you get?
Hint 2
A line’s slope and one point give its equation through point-slope form: $y-y_1=m(x-x_1)$ . Use $A$ as $(x_1,y_1)$ , then pair that equation with $r$ to find where the lines meet.
Hint 3
The intersection $P$ supplies one endpoint, not the answer. Read $T$ from the figure, then take the vertical change over horizontal change from $T$ to $P$ . Did you divide by the full change in $x$ ?

Step-by-step

Graph the intersection, then find the slope
Step 1
Find the slope of the graphed line
Read the marked dots against the axes: $A=(2,5)$ and $B=(6,2)$ . The slope of $s$ is the change in $y$ divided by the change in $x$ . From $A$ to $B$ , subtract in the same order on top and bottom:
$m_s=\frac{2-5}{6-2}$
Type $\frac{2-5}{6-2}$ in Desmos. It displays $-0.75$ ; tap its fraction button:
$m_s=-\frac{3}{4}.$
Step 2
Write an equation for line $s$
Because $A=(2,5)$ lies on $s$ , use point-slope form $y-y_1=m(x-x_1)$ , where $(x_1,y_1)$ is a known point. Put in $A$ and the slope you found to get the equation of $s$ :
$y-5=-\frac{3}{4}(x-2).$
Step 3
Locate $P$ in Desmos
Type $y-5=-\frac{3}{4}(x-2)$ and $2x+y=11$ on separate lines. Click where their graphs cross; Desmos labels the point $(3.6,3.8)$ . An intersection lies on both lines, so $P=(3.6,3.8)$ . That point is not yet the slope the question asks for.
Step 4
Find the slope from $T$ to $P$
Read $T=(1,-2)$ from its dot. From $T$ to $P$ , slope is the change in $y$ over the change in $x$ , not the rise alone. Type $\frac{3.8-(-2)}{3.6-1}$ below the two equations. Desmos displays $2.230769\ldots$ ; tap its fraction button to see $\frac{29}{13}$ . So the slope of $\overline{PT}$ is $\frac{29}{13}$ . Choice A.

Question 298·Hard·Algebra

Strategy

Hints

Step-by-step

Lessons that teach this

Study Hints

Strategy

Hints

Step-by-step

Lessons that teach this