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Mathematics
  1. Given the linear equation y = 2 3 x + 2 y=(2)/(3)x+2, which table of values represents this equation?
    A.
x x y y
-3 0
0 2
3 4
x y -3 0 0 2 3 4
B.
x x y y
0 2
2 5
3 4
x y 0 2 2 5 3 4
C.
x x y y
-3 0
2 0
6 6
x y -3 0 2 0 6 6
D.
x x y y
-9 -6
-6 -2
0 2
x y -9 -6 -6 -2 0 2
  1. What is the y y-intercept of the line that contains the points ( 3 , 3 ) (3,3) and ( 6 , 1 ) (6,-1) ?
    A. 5
    B. 6
    C. 7
    D. 8
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  1. To find the table of values that satisfies the equation y = 2 3 x + 2 y=(2)/(3)x+2, we can substitute different values of x x and solve for y y. Let’s check each table of values:
    A)
    When x = 3 x=-3,
    y = 2 3 ( 3 ) + 2 = 2 y=(2)/(3)(-3)+2=-2
    When x = 0 x=0,
    y = 2 3 ( 0 ) + 2 = 2 y=(2)/(3)(0)+2=2
    When x = 3 x=3,
    y = 2 3 ( 3 ) + 2 = 4 y=(2)/(3)(3)+2=4
    B)
    When x = 0 x=0,
    y = 2 3 ( 0 ) + 2 = 2 y=(2)/(3)(0)+2=2
    When x = 2 x=2,
    y = 2 3 ( 2 ) + 2 = 8 3 y=(2)/(3)(2)+2=(8)/(3)
    When x = 3 x=3,
    y = 2 3 ( 3 ) + 2 = 10 3 y=(2)/(3)(3)+2=(10)/(3)
    C)
    When x = 3 x=-3,
    y = 2 3 ( 3 ) + 2 = 0 y=(2)/(3)(-3)+2=0
    When x = 2 x=2,
    y = 2 3 ( 2 ) + 2 = 8 3 y=(2)/(3)(2)+2=(8)/(3)
    When x = 6 x=6,
    y = 2 3 ( 6 ) + 2 = 6 y=(2)/(3)(6)+2=6
    D)
    When x = 9 x=-9,
    y = 2 3 ( 9 ) + 2 = 4 y=(2)/(3)(-9)+2=-4
    When x = 6 x=-6,
    y = 2 3 ( 6 ) + 2 = 2 y=(2)/(3)(-6)+2=-2
    When x = 0 x=0,
    y = 2 3 ( 0 ) + 2 = 2 y=(2)/(3)(0)+2=2
    Therefore, the table of values that represents the equation y = 2 3 x + 2 y=(2)/(3)x+2 is table (A) (A).
  2. The y y-intercept of a line is the value of y y at which the line intersects the y y-axis (where x = 0 x=0). To find the y y-intercept of the line containing the points ( 3 , 3 ) (3,3) and ( 6 , 1 ) (6,-1), we can use the point-slope formula:
    y y 1 = m ( x x 1 ) y-y_(1)=m(x-x_(1))
    where m m is the slope of the line and ( x 1 , y 1 ) (x_(1),y_(1)) is one of the points on the line.
    We can start by finding the slope:
    m = y 2 y 1 x 2 x 1 = 1 3 6 3 = 4 3 m=(y_(2)-y_(1))/(x_(2)-x_(1))=(-1-3)/(6-3)=-(4)/(3)
    Using the point-slope formula with ( x 1 , y 1 ) = ( 3 , 3 ) (x_(1),y_(1))=(3,3):
    y 3 = 4 3 ( x 3 ) y-3=-(4)/(3)(x-3)
    Simplifying:
    y 3 = 4 3 x + 4 y-3=-(4)/(3)x+4
    y = 4 3 x + 7 y=-(4)/(3)x+7
    The y y-intercept occurs when x = 0 x=0, so we substitute x = 0 x=0 into the equation:
    y = 4 3 ( 0 ) + 7 = 7 y=-(4)/(3)(0)+7=7
    Therefore, the y y-intercept is (C) 7 (C) 7.
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