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Difficulty: MediumLinear Equations in Two Variables

A linear relationship between xx and yy is represented by the values in the table below.

xxyy
−2-2aa
1155
441717
77bb

What is the value of b−ab - a?

  1. A
    2.25
  2. B
    9
  3. 36Answer
  4. D
    22

Answer

36
The constant rate of change (slope) of the relationship is 4, which is found by dividing the difference in yy-values by the difference in xx-values for the given points: 17−54−1=4\frac{17 - 5}{4 - 1} = 4. The value of b−ab - a represents the change in yy over the interval from x=−2x = -2 to x=7x = 7. The length of this interval is 7−(−2)=97 - (-2) = 9. Multiplying the rate of change by this interval length gives the total change in yy: 4×9=364 \times 9 = 36. Alternatively, solving for the equation yields y=4x+1y = 4x + 1, where substituting x=−2x = -2 gives a=−7a = -7 and substituting x=7x = 7 gives b=29b = 29, and their difference is 29−(−7)=3629 - (-7) = 36.

Step-by-Step Solution

1
Find the constant rate of change (slope) of the linear relationship using the points (1,5)(1, 5) and (4,17)(4, 17).
The slope mm is 17−54−1=123=4\frac{17 - 5}{4 - 1} = \frac{12}{3} = 4.
Since the relationship is linear, the rate of change is constant between any two points.
2
Determine the value of aa when x=−2x = -2.
Using the point (1,5)(1, 5) and moving to x=−2x = -2, the change in xx is −3-3. The corresponding change in yy is 4×(−3)=−124 \times (-3) = -12. Thus, a=5−12=−7a = 5 - 12 = -7.
This establishes the value of the first variable in the expression.
3
Determine the value of bb when x=7x = 7.
Using the point (4,17)(4, 17) and moving to x=7x = 7, the change in xx is +3+3. The corresponding change in yy is 4×3=124 \times 3 = 12. Thus, b=17+12=29b = 17 + 12 = 29.
This establishes the value of the second variable in the expression.
4
Calculate the value of b−ab - a.
b−a=29−(−7)=29+7=36b - a = 29 - (-7) = 29 + 7 = 36.
Subtracting a negative number is equivalent to adding its positive counterpart.

Key Concept

Linear rate of change and evaluation of linear relationships from tables of values

Alternative Method

Instead of calculating the individual values of aa and bb, recognize that b−ab - a is the total change in yy over the interval from x=−2x = -2 to x=7x = 7. The change in xx is 7−(−2)=97 - (-2) = 9. Since the constant rate of change (slope) is 44, the change in yy is simply 4×9=364 \times 9 = 36.
Estimated Time:1m 30s
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Topics

Linear Equations in Two Variables

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