12 multiple-choice questions, progressively harder.
The line through (2,k)(2, k)(2,k) and (5,7)(5, 7)(5,7) has slope 222. What is kkk?
Solution
Correct answer: D
Set the slope formula equal to 222 and solve for kkk.
7−k5−2=2,7−k3=2,7−k=6\frac{7 - k}{5 - 2} = 2, \qquad \frac{7 - k}{3} = 2, \qquad 7 - k = 65−27−k=2,37−k=2,7−k=6
So k=1k = 1k=1.
The line through (a,3)(a, 3)(a,3) and (6,9)(6, 9)(6,9) has slope 333. What is aaa?
Correct answer: B
Set the slope formula equal to 333 and solve for aaa.
9−36−a=3,66−a=3,6−a=2\frac{9 - 3}{6 - a} = 3, \qquad \frac{6}{6 - a} = 3, \qquad 6 - a = 26−a9−3=3,6−a6=3,6−a=2
So a=4a = 4a=4.
Which line is the steepest: one with slope 12\tfrac{1}{2}21, one with slope 222, or one with slope −3-3−3?
Correct answer: C
Steepness is the size of the slope, so compare absolute values.
∣12∣=0.5,∣2∣=2,∣−3∣=3\left|\tfrac{1}{2}\right| = 0.5, \qquad |2| = 2, \qquad |-3| = 321=0.5,∣2∣=2,∣−3∣=3
The largest is 333, so the slope −3-3−3 line is steepest.
What is the slope of the line shown on the grid?
Correct answer: A
Count off the slope triangle from the left point. The run is 333 units to the right and the line drops 666 units, a rise of −6-6−6.
m=−63=−2m = \frac{-6}{3} = -2m=3−6=−2
So the slope is −2-2−2.
The points (2,5)(2, 5)(2,5), (4,9)(4, 9)(4,9), and (6,13)(6, 13)(6,13) all lie on one line. What is its slope?
Use any two of the points, since a line has one slope.
m=9−54−2=42=2m = \frac{9 - 5}{4 - 2} = \frac{4}{2} = 2m=4−29−5=24=2
So the slope is 222: each step of 222 in xxx raises yyy by 444.
The line through (−1,4)(-1, 4)(−1,4) and (3,k)(3, k)(3,k) has slope −1-1−1. What is kkk?
Set the slope formula equal to −1-1−1 and solve for kkk.
k−43−(−1)=−1,k−44=−1,k−4=−4\frac{k - 4}{3 - (-1)} = -1, \qquad \frac{k - 4}{4} = -1, \qquad k - 4 = -43−(−1)k−4=−1,4k−4=−1,k−4=−4
So k=0k = 0k=0.
A line has slope 555. Between two of its points the run is 333. What is the rise?
Since slope is rise over run, the rise is slope times run.
rise=m×run=5×3=15\text{rise} = m \times \text{run} = 5 \times 3 = 15rise=m×run=5×3=15
So the rise is 151515.
What is the slope of the line through (−2,−5)(-2, -5)(−2,−5) and (4,7)(4, 7)(4,7)?
Apply the slope formula.
m=7−(−5)4−(−2)=126=2m = \frac{7 - (-5)}{4 - (-2)} = \frac{12}{6} = 2m=4−(−2)7−(−5)=612=2
So the slope is 222.
A line has slope −12-\tfrac{1}{2}−21 and passes through (4,0)(4, 0)(4,0). Which point is also on the line?
Read −12-\tfrac{1}{2}−21 as a run of 222 and a rise of −1-1−1. Step left 222 and up 111 from (4,0)(4, 0)(4,0).
(4,0)⟶(4−2, 0+1)=(2,1)(4, 0) \longrightarrow (4 - 2,\; 0 + 1) = (2, 1)(4,0)⟶(4−2,0+1)=(2,1)
So (2,1)(2, 1)(2,1) is on the line.
What is the slope of the line through (−1,7)(-1, 7)(−1,7) and (2,1)(2, 1)(2,1)?
Apply the slope formula and simplify.
m=1−72−(−1)=−63=−2m = \frac{1 - 7}{2 - (-1)} = \frac{-6}{3} = -2m=2−(−1)1−7=3−6=−2
Which two lines are equally steep?
Two lines are equally steep when their slopes have the same size (absolute value).
∣23∣=∣−23∣=23\left|\tfrac{2}{3}\right| = \left|-\tfrac{2}{3}\right| = \tfrac{2}{3}32=−32=32
So the slopes 23\tfrac{2}{3}32 and −23-\tfrac{2}{3}−32 give equally steep lines, one rising and one falling.
What is the slope of the line through (−4,1)(-4, 1)(−4,1) and (2,−8)(2, -8)(2,−8)?
m=−8−12−(−4)=−96=−32m = \frac{-8 - 1}{2 - (-4)} = \frac{-9}{6} = -\frac{3}{2}m=2−(−4)−8−1=6−9=−23
So the slope is −32-\tfrac{3}{2}−23.
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