Slopes of Perpendicular Lines
In the _xy_ plane, lines _j_ and _k_ are perpendicular at the point (4, 7). Line _j_ has a negative _y_-intercept.
| Quantity A | Quantity B |
|----|----|
| The _y_- intercept of line _k_ | 10 |
Incorrect.
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If you were to sketch line _j_ intersecting the _y_-axis at an extremely negative value, the slope of line _k_ will be very close to 0, making line _k_ almost horizontal. The _y_-intercept will be only slightly higher than 7, the _y_-value at the point of intersection.
Quantity A will not necessarily be greater than Quantity B.
If you were to sketch line _j_ intersecting the _y_-axis at an extremely negative value, the slope of line _k_ will be very close to 0, making line _k_ almost horizontal. The _y_-intercept will be only slightly higher than 7, the _y_-value at the point of intersection.
In this case, Quantity B will still be greater than Quantity A. Since Quantity A must be between these two cases, Quantity B is always greater.
Incorrect.
To see if the two quantities are equal, calculate the slope of both lines and then check if the _y_-intercept of _j_ is negative.
If line _k_ contains the point (0, 10) and (4, 7), then it will have a slope of $$-\frac{3}{4}$$ and line _j_ will have a slope of $$\frac{4}{3}$$.
$$\displaystyle y = \frac{4}{3}x + b$$
$$\displaystyle 7 = \frac{4}{3}(4) + b$$
$$\displaystyle 7 = \frac{16}{3} + b$$
$$\displaystyle 7 - \frac{16}{3} = b$$
Since 7 is greater than $$\frac{16}{3}$$, the _y_-intercept of _j_ would be positive in this case, and that contradicts the given information. The two quantities are not equal.
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Incorrect.
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Although there are a range of possible values for Quantity A, there is enough information to definitively compare the two quantities. It may help to use the extreme cases for Quantity A.
For example, if the largest possible values for the _y_-intercept of _k_ occur as the _y_-intercept of _j_ approaches 0, calculate the case for when the intercept is exactly 0. Quantity A must be less than the _y_-intercept calculated in that extreme case.
Correct.
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Calculate the case for when the _y_-intercept of _j_ is exactly 0. Quantity A must be less than the _y_-intercept of _k_ calculated in that extreme case.
Under this assumption, line _j_ goes through the points (0, 0) and (4, 7). That gives _j_ a slope of $$\frac{7-0}{4-0}=\frac{7}{4}$$. Thus, the line perpendicular to it (_k_) a slope of $$-\frac{4}{7}$$.
If line _k_ has a slope of $$-\frac{4}{7}$$ and a point (4, 7), you can use the slope-intercept equation to solve for the value of the _y_-intercept.
$$\displaystyle y = -\frac{4}{7}x + b$$
$$\displaystyle 7 = -\frac{4}{7}\cdot 4 + b$$
$$\displaystyle 7 = -\frac{16}{7} + b$$
$$\displaystyle 7 + \frac{16}{7} = b$$
$$\displaystyle \frac{49}{7} + \frac{16}{7} = b$$
$$\displaystyle \frac{65}{7} = b$$
$$\displaystyle \frac{65}{7} < 10$$
Under the most extreme case, Quantity A is less than $$\frac{65}{7}$$, which is less than 10, so Quantity A is less than 10 (Quantity B).
Quantity A is greater.
Quantity B is greater.
The two quantities are equal.
The relationship cannot be determined from the information given.
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