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The graph of a line in the xy-plane has a slope of 4 and contains the point (1,-5). The graph of a second line passes through the points (0,4) and (12,0). If the two lines intersect at the point (a,b), what is the value of a-b ? Choose 1 answer: (A) -9 (B) -5 (C) 0 (D) 4

The graph of a line in the xyxy-plane has a slope of 44 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?\newlineChoose 11 answer:\newline(A) −9-9\newline(B) −5-5\newline(C) 00\newline(D) 44

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Full solution

Q. The graph of a line in the xyxy-plane has a slope of 44 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?\newlineChoose 11 answer:\newline(A) −9-9\newline(B) −5-5\newline(C) 00\newline(D) 44
  1. First Line Equation: The first line has a slope of 44 and passes through the point (1,−5)(1, -5). We can use the point-slope form to write the equation of the first line.\newliney−y1=m(x−x1)y - y_1 = m(x - x_1)\newliney−(−5)=4(x−1)y - (-5) = 4(x - 1)
  2. Second Line Equation: Simplify the equation of the first line.\newliney+5=4x−4y + 5 = 4x - 4\newliney=4x−4−5y = 4x - 4 - 5\newliney=4x−9y = 4x - 9\newlineThis is the equation of the first line.
  3. Calculate Second Line Slope: The second line passes through the points (0,4)(0, 4) and (12,0)(12, 0). We can find the slope of the second line using these points.\newlineSlope = (y2−y1)/(x2−x1)(y_2 - y_1) / (x_2 - x_1)\newlineSlope = (0−4)/(12−0)(0 - 4) / (12 - 0)
  4. Write Second Line Equation: Calculate the slope of the second line.\newlineSlope = −412-\frac{4}{12}\newlineSlope = −13-\frac{1}{3}\newlineThis is the slope of the second line.
  5. Find Intersection Point: Now we can write the equation of the second line using the point-slope form and one of the points, for example, (0,4)(0, 4).\newliney−y1=m(x−x1)y - y_1 = m(x - x_1)\newliney−4=(−13)(x−0)y - 4 = \left(-\frac{1}{3}\right)(x - 0)
  6. Solve for xx: Simplify the equation of the second line.y−4=(−13)xy - 4 = \left(-\frac{1}{3}\right)xy=(−13)x+4y = \left(-\frac{1}{3}\right)x + 4This is the equation of the second line.
  7. Calculate x-coordinate: To find the intersection point (a,b)(a, b), we set the equations of the two lines equal to each other.\newline4x−9=(−13)x+44x - 9 = (-\frac{1}{3})x + 4
  8. Calculate y-coordinate: Solve for xx by combining like terms.4x+(13)x=4+94x + (\frac{1}{3})x = 4 + 9(133)x=13(\frac{13}{3})x = 13
  9. Intersection Point Coordinates: Divide both sides by (13/3)(13/3) to solve for xx.x=13(13/3)x = \frac{13}{(13/3)}x=13×(313)x = 13 \times \left(\frac{3}{13}\right)x=3x = 3This is the xx-coordinate of the intersection point.
  10. Calculate a−ba - b: Now we substitute x=3x = 3 into one of the line equations to find the y-coordinate of the intersection point. We can use the first line's equation.\newliney=4x−9y = 4x - 9\newliney=4(3)−9y = 4(3) - 9
  11. Calculate a−ba - b: Now we substitute x=3x = 3 into one of the line equations to find the y-coordinate of the intersection point. We can use the first line's equation.\newliney=4x−9y = 4x - 9\newliney=4(3)−9y = 4(3) - 9Calculate the y-coordinate.\newliney=12−9y = 12 - 9\newliney=3y = 3\newlineThis is the y-coordinate of the intersection point.
  12. Calculate a−ba - b: Now we substitute x=3x = 3 into one of the line equations to find the y-coordinate of the intersection point. We can use the first line's equation.y=4x−9y = 4x - 9y=4(3)−9y = 4(3) - 9Calculate the y-coordinate.y=12−9y = 12 - 9y=3y = 3This is the y-coordinate of the intersection point.Now we have the intersection point (a,b)(a, b) which is (3,3)(3, 3). To find a−ba - b, we subtract the y-coordinate from the x-coordinate.a−b=3−3a - b = 3 - 3
  13. Calculate a−ba - b: Now we substitute x=3x = 3 into one of the line equations to find the y-coordinate of the intersection point. We can use the first line's equation.\newliney=4x−9y = 4x - 9\newliney=4(3)−9y = 4(3) - 9Calculate the y-coordinate.\newliney=12−9y = 12 - 9\newliney=3y = 3\newlineThis is the y-coordinate of the intersection point.Now we have the intersection point (a,b)(a, b) which is (3,3)(3, 3). To find a−ba - b, we subtract the y-coordinate from the x-coordinate.\newlinea−b=3−3a - b = 3 - 3Calculate a−ba - b.\newlinea−b=0a - b = 0\newlineThis is the value of a−ba - b.

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