12 multiple-choice questions, progressively harder.
Convert 4x−3y=94x - 3y = 94x−3y=9 to slope-intercept form.
Solution
Correct answer: D
Isolate the yyy-term, then divide every term by −3-3−3.
−3y=−4x+9,y=43x−3-3y = -4x + 9, \qquad y = \tfrac{4}{3}x - 3−3y=−4x+9,y=34x−3
Dividing by −3-3−3 flips both signs, giving slope 43\tfrac{4}{3}34 and intercept (0,−3)(0, -3)(0,−3).
Convert y−5=−14(x+8)y - 5 = -\tfrac{1}{4}(x + 8)y−5=−41(x+8) to slope-intercept form.
Correct answer: B
Distribute the slope, then add 555 to both sides.
y−5=−14x−2,y=−14x+3y - 5 = -\tfrac{1}{4}x - 2, \qquad y = -\tfrac{1}{4}x + 3y−5=−41x−2,y=−41x+3
Here −14(8)=−2-\tfrac{1}{4}(8) = -2−41(8)=−2, and −2+5=3-2 + 5 = 3−2+5=3, so the intercept is (0,3)(0, 3)(0,3).
Convert y+2=23(x−6)y + 2 = \tfrac{2}{3}(x - 6)y+2=32(x−6) to slope-intercept form.
Correct answer: C
Distribute the slope, then subtract 222 from both sides.
y+2=23x−4,y=23x−6y + 2 = \tfrac{2}{3}x - 4, \qquad y = \tfrac{2}{3}x - 6y+2=32x−4,y=32x−6
Here 23(−6)=−4\tfrac{2}{3}(-6) = -432(−6)=−4, and −4−2=−6-4 - 2 = -6−4−2=−6, so the intercept is (0,−6)(0, -6)(0,−6).
A line has slope 43\tfrac{4}{3}34 and passes through (3,1)(3, 1)(3,1). What is its yyy-intercept?
Correct answer: A
Write y=43x+by = \tfrac{4}{3}x + by=34x+b and use the point (3,1)(3, 1)(3,1) to find bbb.
1=43(3)+b,1=4+b,b=−31 = \tfrac{4}{3}(3) + b, \qquad 1 = 4 + b, \qquad b = -31=34(3)+b,1=4+b,b=−3
So the yyy-intercept is (0,−3)(0, -3)(0,−3).
The point (k,5)(k, 5)(k,5) lies on the line y=2x−3y = 2x - 3y=2x−3. What is kkk?
Substitute y=5y = 5y=5 and solve for x=kx = kx=k.
5=2k−3,2k=8,k=45 = 2k - 3, \qquad 2k = 8, \qquad k = 45=2k−3,2k=8,k=4
So k=4k = 4k=4, and the point is (4,5)(4, 5)(4,5).
Convert 12x+13y=1\tfrac{1}{2}x + \tfrac{1}{3}y = 121x+31y=1 to slope-intercept form.
Isolate the yyy-term, then multiply every term by 333 to clear the fraction on yyy.
13y=−12x+1,y=−32x+3\tfrac{1}{3}y = -\tfrac{1}{2}x + 1, \qquad y = -\tfrac{3}{2}x + 331y=−21x+1,y=−23x+3
So the slope is −32-\tfrac{3}{2}−23 and the yyy-intercept is (0,3)(0, 3)(0,3).
What is the yyy-intercept of the line y−4=12(x+6)y - 4 = \tfrac{1}{2}(x + 6)y−4=21(x+6)?
Convert to slope-intercept form by distributing and adding 444.
y−4=12x+3,y=12x+7y - 4 = \tfrac{1}{2}x + 3, \qquad y = \tfrac{1}{2}x + 7y−4=21x+3,y=21x+7
Since 12(6)=3\tfrac{1}{2}(6) = 321(6)=3 and 3+4=73 + 4 = 73+4=7, the yyy-intercept is (0,7)(0, 7)(0,7).
Which statement best describes the graph of y=34x−2y = \tfrac{3}{4}x - 2y=43x−2?
The slope 34\tfrac{3}{4}43 is positive, so the line rises to the right, and the constant −2-2−2 sets the crossing at (0,−2)(0, -2)(0,−2).
m=34>0,b=−2m = \tfrac{3}{4} > 0, \qquad b = -2m=43>0,b=−2
So the line rises to the right and crosses the yyy-axis at (0,−2)(0, -2)(0,−2).
A line falls as you move to the right and crosses the yyy-axis below the origin. Which could be its equation?
Falling to the right means the slope is negative (m<0m < 0m<0), and crossing below the origin means the intercept is negative (b<0b < 0b<0).
m<0 and b<0 ⇒ y=−2x−3m < 0 \text{ and } b < 0 \;\Rightarrow\; y = -2x - 3m<0 and b<0⇒y=−2x−3
Only y=−2x−3y = -2x - 3y=−2x−3 has both a negative slope and a negative yyy-intercept.
Convert 5x−10y=205x - 10y = 205x−10y=20 to slope-intercept form.
Isolate the yyy-term, then divide every term by −10-10−10.
−10y=−5x+20,y=12x−2-10y = -5x + 20, \qquad y = \tfrac{1}{2}x - 2−10y=−5x+20,y=21x−2
Dividing −5x-5x−5x by −10-10−10 gives 12x\tfrac{1}{2}x21x, and 20÷(−10)=−220 \div (-10) = -220÷(−10)=−2, so the intercept is (0,−2)(0, -2)(0,−2).
Convert −x+2y=8-x + 2y = 8−x+2y=8 to slope-intercept form.
Isolate the yyy-term, then divide every term by 222.
2y=x+8,y=12x+42y = x + 8, \qquad y = \tfrac{1}{2}x + 42y=x+8,y=21x+4
So the slope is 12\tfrac{1}{2}21 and the yyy-intercept is (0,4)(0, 4)(0,4).
What are both intercepts of the line y=23x−4y = \tfrac{2}{3}x - 4y=32x−4?
The yyy-intercept is the constant, (0,−4)(0, -4)(0,−4). For the xxx-intercept, set y=0y = 0y=0.
0=23x−4,23x=4,x=60 = \tfrac{2}{3}x - 4, \qquad \tfrac{2}{3}x = 4, \qquad x = 60=32x−4,32x=4,x=6
So the xxx-intercept is (6,0)(6, 0)(6,0) and the yyy-intercept is (0,−4)(0, -4)(0,−4).
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