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The graph of a line in the xy-plane has a slope of 1 and contains the point (3,0). The graph of a second line has a slope of -(1)/(4) and contains the point (-7,0). If the two lines intersect at the point (a,b), what is the value of a+b ?

The graph of a line in the xy x y -plane has a slope of 11 and contains the point (3,0) (3,0) . The graph of a second line has a slope of −14 -\frac{1}{4} and contains the point (−7,0) (-7,0) . If the two lines intersect at the point (a,b) (a, b) , what is the value of a+b a+b ?

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Full solution

Q. The graph of a line in the xy x y -plane has a slope of 11 and contains the point (3,0) (3,0) . The graph of a second line has a slope of −14 -\frac{1}{4} and contains the point (−7,0) (-7,0) . If the two lines intersect at the point (a,b) (a, b) , what is the value of a+b a+b ?
  1. First line equation: First line equation using point-slope form: y−y1=m(x−x1)y - y_1 = m(x - x_1), where mm is slope and (x1,y1)(x_1, y_1) is the point.\newlineFor the first line: y−0=1(x−3)y - 0 = 1(x - 3), which simplifies to y=x−3y = x - 3.
  2. Second line equation: Second line equation using point-slope form: y−y1=m(x−x1)y - y_1 = m(x - x_1). For the second line: y−0=−14(x+7)y - 0 = -\frac{1}{4}(x + 7), which simplifies to y=−14x−74y = -\frac{1}{4}x - \frac{7}{4}.
  3. Find intersection point: Find the intersection point by setting the two equations equal to each other: x−3=−14x−74x - 3 = -\frac{1}{4}x - \frac{7}{4}.
  4. Solve for x: Solve for x: x+(14)x=3−(74)x + \left(\frac{1}{4}\right)x = 3 - \left(\frac{7}{4}\right). This simplifies to (54)x=54\left(\frac{5}{4}\right)x = \frac{5}{4}.
  5. Substitute xx into first line: Divide both sides by (5/4)(5/4) to get x=1x = 1.
  6. Solve for yy: Substitute x=1x = 1 into the first line's equation to find yy: y=1−3y = 1 - 3.
  7. Intersection point: Solve for yy: y=−2y = -2.
  8. Calculate a+ba + b: The intersection point is (1,−2)(1, -2), so a=1a = 1 and b=−2b = -2.
  9. Calculate a+ba + b: The intersection point is (1,−2)(1, -2), so a=1a = 1 and b=−2b = -2.Calculate a+ba + b: 1+(−2)=−11 + (-2) = -1.

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