Solutions

  1. (0, 7)

    A line intersects the y-axis at the y-intercept. Because this equation is written in slope-intercept form, the y-intercept is easy to identify: 7. Thus, the line intersects the y-axis at the point (0, 7).

  2. (−20, 0)

    A line intersects the x-axis at the x-intercept, or when the y-coordinate is equal to 0. Substitute 0 for y and solve for x:

    x = 0 − 20
    x = −20

  3. 7

    Substitute the coordinates (3, 2) for x and y and solve for z:

    3 = −2(2) + z
    3 = −4 + z
    z = 7

  4. 6

    Substitute the coordinates (−3, 0) for x and y and solve for z:

      0 = z(−3) + 18
    3z = 18
      z = 6

  5. (0, 4)

    Use the information given to find the equation of the line:

    The variable b represents the y-intercept. Therefore, the line intersects the y-axis at (0, 4).

  6. II

    First, rewrite the line in slope-intercept form:

    y = x − 18

    Find the intercepts by setting x to 0 and y to 0:

    y = 0 − 18
    y = −18    		 0 = x − 18
    x = 18

    Plot the points: (0, −18), and (18, 0). From the sketch, you can see that the line does not pass through quadrant II.

  7. II and IV

    First, rewrite the line in slope-intercept form:

    Notice from the equation that the y-intercept of the line is (0,0). This means that the line crosses the y-intercept at the origin, so the x- and y-intercepts are the same. To find another point on the line, substitute any convenient number for x; in this case, 10 would be a convenient, or “smart,” number:

    The point (10, 1) is on the line.

    Plot the points: (0, 0) and (10, 1). From the sketch, you can see that the line does not pass through quadrants II and IV.

  8. I, II, and III

    The line is already written in slope-intercept form:

    Find the intercepts by setting x to 0 and y to 0:

    Plot the points: (−1,000,000,000, 0) and (0, 1,000,000). From the sketch, which is obviously not to scale, you can see that the line passes through quadrants I, II, and III.

  9. I, II, and III

    First, rewrite the line in slope-intercept form:

    Find the intercepts by setting x to 0 and y to 0:

    Plot the points: (−18, 0) and (0, 9). From the sketch, you can see that the line passes through quadrants I, II, and III.

  10. First, calculate the slope of the line:

    You can see from the graph that the line crosses the y-axis at (0, 6). The equation of the line is:

  11. (5, −3)

    To find the coordinates of the point of intersection, solve the system of two linear equations. You could turn both equations into slope-intercept form and set them equal to each other, but it is easier to multiply the first equation by 2 and then add the second equation:

    2x + y = 7
    4x + 2y = 14
    3x − 2y = 21    		 (first equation)
    (first equation multipied by 2)
    (second equation)    		 7x = 35
      x = 5    		 (sum of previous two eqations)
     
     

    Now plug x = 5 into either equation:

    2x + y = 7
    2(5) + y = 7    		 (first equation)
         		 10 + y = 7
    y = −3

    Thus, the point (5, −3) is the point of intersection. There is no need to graph the two lines to find the point of intersection.

  12. (B)

    The line is illustrated on the coordinate plane shown. Because the equation is already in slope intercept form (y = mx + b), you can read the y-intercept directly from the b position, and use the slope to determine the x-intercept. A slope of 3/2 means that the line corresponding to this equation will rise 3 for every 2 that it runs. You don’t need to determine the exact x-intercept to see that it is positive, and so greater than −2.

    Alternatively, you could set each variable equal to 0 and determine the intercepts.

    To determine the y-intercept, set x equal to 0, then solve for y:

    To determine the x-intercept, set y equal to 0, then solve for x :

    Quantity A Quantity B
    The y-intercept of the line
    −2    		 The x-intercept of the line

    Therefore, Quantity B is greater.

  13. (A)

    The best method would be to put each equation into slope-intercept form (y = mx + b) and see which has the greater value for m, which represents the slope. Start with the equation in Quantity A:

    Quantity A Quantity B
    The slope of the line
    2x + 5y = 10 is     		The slope of the line
    5x + 2y = 10

    Now find the slope of the equation in Quantity B:

    Quantity A Quantity B
    The slope of the line
    5x + 2y = 10 is

    Be careful. Remember that . Therefore, Quantity A is greater.

  14. (C)

    The illustration shows the two points from Quantity A, here labeled A1 and A2, and the two points from Quantity B, here labeled B1 and B2. You can construct a right triangle from the A values by finding a point (0, 0) directly below A1 and directly to the right of A2. This right triangle has legs of 2 (the change from −2 to 0) and 9 (the change from 0 to 9). You can plug those values into the Pythagorean theorem and solve for the hypotenuse:

    Quantity A Quantity B
    The distance between points
    (0, 9) and     		The distance between points (3, 9)
    and (10, 3)

    You can construct a right triangle from the B values by finding a point (3, 3) directly below B1 and directly to the left of B2. This right triangle has legs of 7 (the change from 3 to 10) and 6 (the change from 3 to 9). You can plug those values into the Pythagorean theorem and solve for the hypotenuse:

    Quantity A Quantity B
    The distance between points
    (0, 9) and     		The distance between points (3, 9)
    and

    Therefore, the two quantities are equal.