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Page Two Here's an example: Find the equation of the line that passes through (1, 3) and (4, 15) we can work oit the slope: m = Δy over Δx = (15 - 3) over (4 - 1) = 12 over 3 = 4 But now we're stuck. We don't know the y intercept. This requires a different method. It's called the point-slope method. It's a formula you can use tho get the equation. You need to know the slope and one point on the line.
Find the equation of the line that passes through (1, 3) and (4, 15) We worked out the slope. It's 4. Let's use (1, 3) as the point. Which one we choose doesn't matter. m = 4 (x1, y1) = (1, 3) (y - y1) = m(x - x1) (y - 3) = 4(x - 1) Now simplify and solve for y y - 3 = 4x - 4 y = 4x -1 Let's try another one: Find the equation of the line that passes through (-2, 5) and (1, 14) and put it in standard form We'll need the slope first: m = Δy over Δx = (14 - 5) over (1 - -2) = 9 over 3 = 3 so we have m = 3 and using point (1, 14): (y - y1) = m(x - x1) (y - 14) = 3(x - 1) Now simplify and put in standard form y - 14 = 3x - 3 -3x + y - 11 = 0 3x - y + 11 = 0 Suggestion: When choosing one of the points for (x1, y1), pick the simplest one, with small numbers and/or no negatives if possible. This will make the algebra a little easier. This method also works if you are given the slope and a point that's not the y intercept: Find the equation of the line with slope -2 and passing through the point (4, -6) We have m = -2 and using point (4, -6): (y - y1) = m(x - x1) (y - -6) = -2(x - 4) y + 6 = -2x + 8 y = -2x + 2 or 2x + y - 2 = 0 The last example: Find the equation of the line in standard form that is perendicular to 4x - 3y - 12 = 0 and passes through (8, 4) Notice that we don't know the slope or the y intercept. But we have a hint about the slope: 4x - 3y - 12 = 0 solve for y: -3y = -4x + 12 y = 4/3x - 4 This line has slope 4/3, so the line we want has slope -3/4 Now we know m = -3/4 and (x1, y1) = (8, 4) (y - y1) = m(x - x1) (y - 4) = -3/4(x - 8) y - 4 = -3/4x + 6 y = -3/4x + 10 in slope-intercept form 3/4x + y - 10 = 0 now multiply all terms by 4 to remove fractions: 3x + 4y - 40 = 0 in standard form Linear Equations | Parallel and Perpendicular | Equation Forms | Finding Equations |