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The graph of a line in the xyxy-plane has a slope of and contains the point (1,−5)(1,-5). The graph of a second line passes through the points (0,4)(0,4) and (12,0)(12,0). If the two lines intersect at the point (a,b)(a,b) what is the value of a−ba-b?

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Full solution

Q. The graph of a line in the xyxy-plane has a slope of and contains the point (1,−5)(1,-5). The graph of a second line passes through the points (0,4)(0,4) and (12,0)(12,0). If the two lines intersect at the point (a,b)(a,b) what is the value of a−ba-b?
  1. Find First Line Equation: First, let's find the equation of the first line with the given slope mm and point (1,−5)(1, -5). The slope-intercept form of a line is y=mx+by = mx + b, where mm is the slope and bb is the y-intercept. We are given the slope m=(slope not provided in the problem, assuming it’s a typo and should be a number)m = (\text{slope not provided in the problem, assuming it's a typo and should be a number}), and we have the point (1,−5)(1, -5). Let's plug in the values into the equation y=mx+by = mx + b to find bb. −5=m×1+b-5 = m \times 1 + b (1,−5)(1, -5)00 (1,−5)(1, -5)11
  2. Find Second Line Equation: Now, let's find the equation of the second line that passes through the points (0,4)(0, 4) and (12,0)(12, 0). The slope mm of a line passing through two points (x1,y1)(x_1, y_1) and (x2,y2)(x_2, y_2) is given by m=y2−y1x2−x1m = \frac{y_2 - y_1}{x_2 - x_1}. For the points (0,4)(0, 4) and (12,0)(12, 0), the slope mm is: m=0−412−0m = \frac{0 - 4}{12 - 0} (12,0)(12, 0)00 (12,0)(12, 0)11
  3. Intersection Point Calculation: Since the second line passes through the point (0,4)(0, 4), which is the y-intercept, the equation of the second line is:\newliney=(−13)x+4y = \left(-\frac{1}{3}\right)x + 4
  4. Intersection Point Calculation: Since the second line passes through the point (0,4)(0, 4), which is the yy-intercept, the equation of the second line is:\newliney=(−13)x+4y = (-\frac{1}{3})x + 4To find the intersection point (a,b)(a, b) of the two lines, we need to set their equations equal to each other.\newlineHowever, we have an issue: the slope of the first line was not provided in the problem statement. Without this information, we cannot find the equation of the first line and therefore cannot find the intersection point.\newlineThis is a critical piece of information missing, and we cannot proceed without it.

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