If the graphs of the functions y = x and y = x² + 3x + 1 intersect at the point (w, u), what is the equation of the line that contains the point (w, u) and that is perpendicular to the line y = (1/2)x – 2?
A. y = -2x - 3 B. y = -2x + 3 C. y = 2x + 3 D. y = 2x - 3 E. y = 3x - 2
(A) The first step is to determine the point (w, u). To accomplish this we set the two equations equal to each other in order to eliminate the y parameter (if they are both equal to y, they are both equal to each other):
x² + 3x + 1 = x.
Now we move all the terms to the left hand side: x² + 2x + 1 = 0. Notice that the left hand side of this equation can be factored to (x + 1)² and equation rewritten as:
(x + 1)² = 0.
It should be clear that the only value of x that will satisfy the equation is x = -1. To determine the y coordinate we simply substitute x = -1 into either of the two given equations to get a value for y (you can convince yourself that it doesn't matter which one we plug into by testing x = -1 in both). The easiest route is to substitute x = -1 into the first equation (y = x) to get y = -1. The point (w, u) therefore, is the point (-1, -1).
To determine the answer there is one more set of operations we must compute. Recall that the product of the slopes of perpendicular lines is equal to -1. That means that the equation of the line we are asked to find must have a slope of -2 (since line y = (1/2)x – 2 has a slope of 1/2. Using the well known point-slope equation for lines we set up the ratio (y+1)/(x+1) = -2/1 and cross multiply before solving for y: y+1 = -2x -2, which implies: y = -2x - 3, or (A), the answer to our question.
--------------------------------- could use a bunch of graphics for the explanation built into the page. Thanks.
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