12 multiple-choice questions, progressively harder.
Convert 3x+4y=203x + 4y = 203x+4y=20 to slope-intercept form.
Solution
Correct answer: B
Isolate the yyy-term, then divide every term by 444.
4y=−3x+20,y=−34x+54y = -3x + 20, \qquad y = -\tfrac{3}{4}x + 54y=−3x+20,y=−43x+5
Dividing by 444 reaches every term, giving slope −34-\tfrac{3}{4}−43 and intercept (0,5)(0, 5)(0,5).
What is the slope of the line 6x−2y=76x - 2y = 76x−2y=7?
Correct answer: D
Solve for yyy to expose the slope.
−2y=−6x+7,y=3x−72-2y = -6x + 7, \qquad y = 3x - \tfrac{7}{2}−2y=−6x+7,y=3x−27
So the slope is 333; the −72-\tfrac{7}{2}−27 is the yyy-intercept.
The line shown passes through the two marked points. Which equation represents it?
Correct answer: A
The line crosses the yyy-axis at (0,3)(0, 3)(0,3), so b=3b = 3b=3, and from (0,3)(0, 3)(0,3) to (2,1)(2, 1)(2,1) it drops 222 over a run of 222.
m=1−32−0=−22=−1,y=−x+3m = \frac{1 - 3}{2 - 0} = \frac{-2}{2} = -1, \qquad y = -x + 3m=2−01−3=2−2=−1,y=−x+3
Only y=−x+3y = -x + 3y=−x+3 passes through both marked points.
What is the xxx-intercept of the line y=34x−6y = \tfrac{3}{4}x - 6y=43x−6?
Set y=0y = 0y=0 and solve for xxx.
0=34x−6,34x=6,x=80 = \tfrac{3}{4}x - 6, \qquad \tfrac{3}{4}x = 6, \qquad x = 80=43x−6,43x=6,x=8
So the xxx-intercept is (8,0)(8, 0)(8,0); (0,−6)(0, -6)(0,−6) is the yyy-intercept.
Which of these lines has yyy-intercept (0,0)(0, 0)(0,0)?
Correct answer: C
A yyy-intercept of (0,0)(0, 0)(0,0) means the constant is 000, so the line passes through the origin.
y=−6x=−6x+0 ⇒ (0,0)y = -6x = -6x + 0 \;\Rightarrow\; (0, 0)y=−6x=−6x+0⇒(0,0)
So y=−6xy = -6xy=−6x has yyy-intercept (0,0)(0, 0)(0,0); y=x−4y = x - 4y=x−4 crosses at (0,−4)(0, -4)(0,−4).
What are both intercepts of the line 2x−5y=102x - 5y = 102x−5y=10?
Set y=0y = 0y=0 for the xxx-intercept, then x=0x = 0x=0 for the yyy-intercept.
2x=10⇒x=5;−5y=10⇒y=−22x = 10 \Rightarrow x = 5; \qquad -5y = 10 \Rightarrow y = -22x=10⇒x=5;−5y=10⇒y=−2
So the xxx-intercept is (5,0)(5, 0)(5,0) and the yyy-intercept is (0,−2)(0, -2)(0,−2).
A horizontal line passes through (−3,4)(-3, 4)(−3,4). What is its equation?
A horizontal line has slope 000 and fixes yyy at the point's height.
y=4y = 4y=4
Every point on it has height 444, whatever its xxx-value; x=−3x = -3x=−3 would be the vertical line through the point.
What is the xxx-intercept of the line y=−23x+8y = -\tfrac{2}{3}x + 8y=−32x+8?
0=−23x+8,23x=8,x=120 = -\tfrac{2}{3}x + 8, \qquad \tfrac{2}{3}x = 8, \qquad x = 120=−32x+8,32x=8,x=12
So the xxx-intercept is (12,0)(12, 0)(12,0); (0,8)(0, 8)(0,8) is the yyy-intercept.
A line has slope −3-3−3 and the same yyy-intercept as the line y=4x+7y = 4x + 7y=4x+7. What is its equation?
The line y=4x+7y = 4x + 7y=4x+7 has yyy-intercept (0,7)(0, 7)(0,7), so b=7b = 7b=7, and the new slope is −3-3−3.
y=−3x+7y = -3x + 7y=−3x+7
So the equation is y=−3x+7y = -3x + 7y=−3x+7; it shares the intercept 777 but not the slope.
What is the slope of the line through (−2,3)(-2, 3)(−2,3) and (4,3)(4, 3)(4,3), and can it be written as y=mx+by = mx + by=mx+b?
Both points have height 333, so the rise is 000 and the line is horizontal.
m=3−34−(−2)=06=0,y=0x+3=3m = \frac{3 - 3}{4 - (-2)} = \frac{0}{6} = 0, \qquad y = 0x + 3 = 3m=4−(−2)3−3=60=0,y=0x+3=3
A slope of 000 is defined, so the line can be written y=3y = 3y=3; only vertical lines lack a slope-intercept form.
What is the yyy-intercept of the line 4x+3y=−94x + 3y = -94x+3y=−9?
Set x=0x = 0x=0 and solve for yyy.
4(0)+3y=−9,3y=−9,y=−34(0) + 3y = -9, \qquad 3y = -9, \qquad y = -34(0)+3y=−9,3y=−9,y=−3
So the yyy-intercept is (0,−3)(0, -3)(0,−3).
A line crosses the xxx-axis at (6,0)(6, 0)(6,0) and the yyy-axis at (0,4)(0, 4)(0,4). What is its slope?
Use the two intercept points in the slope formula.
m=4−00−6=4−6=−23m = \frac{4 - 0}{0 - 6} = \frac{4}{-6} = -\tfrac{2}{3}m=0−64−0=−64=−32
So the slope is −23-\tfrac{2}{3}−32: the line falls from the yyy-axis down to the xxx-axis.
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