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Find the slope-intercept form for the line passing through (5,7)(5,7) and parallel to the line passing through (3,3)(3,3) and (−9,1)(-9,1).

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Full solution

Q. Find the slope-intercept form for the line passing through (5,7)(5,7) and parallel to the line passing through (3,3)(3,3) and (−9,1)(-9,1).
  1. Calculate Slope: First, we need to find the slope of the line that passes through the points (3,3)(3,3) and (−9,1)(-9,1). 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 the formula m=y2−y1x2−x1m = \frac{y_2 - y_1}{x_2 - x_1}.
  2. Find Parallel Line Slope: Using the points (3,3)(3,3) and (−9,1)(-9,1), we calculate the slope as follows:\newlinem=(1−3)/(−9−3)m = (1 - 3) / (-9 - 3)\newlinem=(−2)/(−12)m = (-2) / (-12)\newlinem=1/6m = 1/6\newlineThe slope of the line passing through (3,3)(3,3) and (−9,1)(-9,1) is 1/61/6.
  3. Determine Y-Intercept: Since the line we are looking for is parallel to the line passing through (3,3)(3,3) and (−9,1)(-9,1), it will have the same slope. Therefore, the slope of our line is also 16\frac{1}{6}.
  4. Substitute Values: Now, we need to find the y-intercept bb of our line. 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 have a point (5,7)(5,7) that lies on our line and the slope m=16m = \frac{1}{6}. We can substitute these values into the slope-intercept form to solve for bb.
  5. Solve for bb: Substituting the point (5,7)(5,7) and the slope 16\frac{1}{6} into the equation y=mx+by = mx + b, we get:\newline7=(16)⋅5+b7 = \left(\frac{1}{6}\right)\cdot 5 + b
  6. Common Denominator: To find bb, we solve the equation:\newline7=56+b7 = \frac{5}{6} + b\newlineTo isolate bb, we subtract 56\frac{5}{6} from both sides of the equation:\newline7−56=b7 - \frac{5}{6} = b
  7. Subtract Numerators: To subtract 56\frac{5}{6} from 77, we need a common denominator. We can write 77 as 426\frac{42}{6}:\newline426−56=b\frac{42}{6} - \frac{5}{6} = b
  8. Calculate Y-Intercept: Now we subtract the numerators:\newline(42−5)/6=b(42 - 5) / 6 = b\newline37/6=b37 / 6 = b\newlineThe y-intercept of our line is 37/637/6.
  9. Final Slope-Intercept Form: We now have both the slope and the y-intercept for our line. The slope-intercept form of the line is: y=16x+376y = \frac{1}{6}x + \frac{37}{6}

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