Answer:
a. (-2,-1)
b. y = -(1/2)x-2
Explanation:
We will look for an equation of the form y=mx+c, where m is the slope and b is the y-intercept (the value of y when x = 0).
Put the two given points into a table to make it easier to calculate the slopes for Lines A and B. See the attachment for the table. Slope is the change in y for a change in x, often termed the "Rise/Run." Using Line A as the example, the change in y (the Rise) is obtained by taking the Point 2 value for y (7) and subtracting the Point 1 value for y (3): Rise = (7 - 3) = 4. The Run is obtained by taking the Point 2 value for x (2) and subracting the Point 1 value of x (0). The ratio of Rise/Run is 4/2, so the slope is 2. The same process is used for Line B to obtain a slope of 3 for that line.
The two equations can be written with these slopes:
Line A: y = 2x + c
Line B: y = 3x + c
We need to find a value of c for each equation that will force the line through either of their two given points. c can be determined by entering either of that line's points into the equation(s) above and solving for c:
Line A:
y = 2x + c
3 = 2*(0) + c for point (0,3)
c = 3
The equation for Line A is y = 2x + 3
Line B:
y = 3x + c
2 = 3*(-1) + c for point (-1,2)
c = 5
The equation for Line B is y = 3x + 5
The graph of these lines is attached. They intersect at point (-2,-1).
[As a note, the solution may also be found by using substitution and solving for both x and y:
y = 2x+3
y = 3x+5
The solution exists when both y's and x's are equal:
Assume y is equal:
2x+3 = 3x+5
-x = 2; x = -2
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Assume x is equal:
Rearrange:
y = 2x+3
2x = y-3
x = (y-3)/2
Use this definition of x in the second line equation:
y = 3x+5
y = 3((y-3)/2)+5
y = (3y - 9)/2 + 5
2y = 3y - 9 + 10
-y = 1
y = -1
The solution is (-2,-1), the same as was found by graphing.]
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A perpendicular line has a slope that is the negative inverse of the referenced line. For Line A, the perpendicual line would be -(1/2). The equation for this line would take the form y = -(1/2)x + c. We want this perpendicular line to intersect the same point that is the solution to the first problem: (-2,-1). Enter this point to find c:
y = -(1/2)x + c
-1 = -(1/2)*(-2) + c for (-2,-1)
-1 = 1+ c
c = -2
The perpendicular line is y = -(1/2)x - 2
See the attached graph.