# Determine the slope of lines a and h

by Lisa

Parallel, Perpendicular, or Intersecting Lines?

Given line “A” with points A (4,4) and B(8,10) and line “H” with points D(-2,3)
and E (0,6),

Determine the slope of line “A” and “H”.

Write the equation of line “A” and “H”.

Are the two lines parallel, perpendicular, or intersecting, at a non-right angle?

### Comments for Determine the slope of lines a and h

 Dec 23, 2012 Parallel, Perpendicular, or Intersecting Lines by: Staff Answer Part I Determine the slope of line “A”. slope = "rise" over "run", or "rise" DIVIDED BY "run" slope = (change in "y" values)/(change in "x" values) slope = Δy / Δx slope = (y₂ - y₁)/(x₂ - x₁) The x-y coordinates for the two points for Line “A” are (4, 4) and (8, 10). These points are already listed in the proper order (left to right) They are listed properly as (4, 4) and (8, 10) because the slope will be calculated as the change of "y" when x increases. In this case "x" will increase from 4 to 8 (left to right). It is worth plotting the two points so the position of each can be visualized. --------------------------------------------

 Dec 23, 2012 Parallel, Perpendicular, or Intersecting Lines by: Staff -------------------------------------------- Part II Calculation of the slope for Line “A” Points: Left and Right (4, 4) and (8, 10) (x₁, y₁) and (x₂,y₂) x₁ = 4 x₂ = 8 y₁ = 4 y₂ = 10 slope = (y₂ - y₁)/(x₂ - x₁) slope = (10 - 4)/(8 - 4) slope = (6)/(4) slope = 6/4 slope = 3/2                  Line “A” slope = 3/2 Write the equation of line “A”. The Point-Slope Form ```Point-Slope Form: (y - y₁) = m (x - x₁) This format uses a single known point (x₁,y₁) and the slope m (which is also a known value) From the problem statement: m (the slope) = 3/2 single known point (x₁,y₁) = (4,4) x₁ = 4 y₁ = 4 (y - y₁) = m (x - x₁) (y - 4) = (3/2)*(x - 4) Point Slope Form of Equation for Line “A”: (y - 4) = (3/2)*(x - 4) ``` The Slope Intercept Form ```Slope Intercept Form y = mx + b (y - 4) = (3/2)*(x - 4) 2*(y - 4) = 2*(3/2)*(x - 4) 2y - 8 = 3x - 8 2y - 8 + 8 = 3x - 12 + 8 2y + 0 = 3x - 4 2y = 3x - 4 2y / 2 = (3x - 4) / 2 y * (2 / 2) = (3x - 4) / 2 y * (1) = (3x - 4) / 2 y = (3x - 4) / 2 y = (3/2)x - 2 Slope Form of Equation for Line “A”: y = (3/2)x - 2 ``` The Standard Form ```Standard Form Ax + By = C A, B, and C are constants y = (3/2)x - 2 (3/2)x - 2 = y (3/2)x - 2 + 2 = y + 2 (3/2)x + 0 = y + 2 (3/2)x - y = y + 2 - y (3/2)x - y = y - y + 2 (3/2)x - y = 0 + 2 (3/2)x - y = 2 Standard Form of Equation for Line “A”: (3/2)x - y = 2 ``` --------------------------------------------

 Dec 23, 2012 Parallel, Perpendicular, or Intersecting Lines by: Staff -------------------------------------------- Part III Determine the slope of line “H”. slope = "rise" over "run", or "rise" DIVIDED BY "run" slope = (change in "y" values)/(change in "x" values) slope = Δy / Δx slope = (y₂ - y₁)/(x₂ - x₁) The x-y coordinates for the two points for Line “H” are (-2, 3) and (0, 6). These points are also listed in the proper order (left to right) They are listed properly as (-2, 3) and (0, 6). The slope for Line “H” will be calculated in exactly the same way as the slope for Line “A” was calculated: the change of "y" when x increases. In this case "x" will increase from -2 to 3 (left to right). The two points are plotted below. --------------------------------------------

 Dec 23, 2012 Parallel, Perpendicular, or Intersecting Lines by: Staff --------------------------------------------Part IVCalculation of the slope for Line “H” Points: Left and Right(-2, 3) and (0, 6)(x₁, y₁) and (x₂,y₂)x₁ = -2x₂ = 0y₁ = 3y₂ = 6slope = (y₂ - y₁)/(x₂ - x₁)slope = (6 - 3)/(0 - (-2))slope = (3)/(2)slope = 3/2                 Line “H” slope = 3/2 Write the equation of line “H”.The Point-Slope Form `Point-Slope Form: (y - y₁) = m (x - x₁) This format uses a single known point (x₁,y₁) and the slope m (which is also a known value)From the problem statement: m (the slope) = 3/2 single known point (x₁,y₁) = (-2,3) x₁ = -2 y₁ = 3(y - y₁) = m (x - x₁)(y - 3) = (3/2)*(x - (-2)) (y - 3) = (3/2)*(x + 2) Point Slope Form of Equation for Line “H”: (y - 3) = (3/2)*(x + 2) ` The Slope Intercept Form `Slope Intercept Formy = mx + b(y - 3) = (3/2)*(x + 2)2*(y - 3) = 2*(3/2)*(x + 2) 2y - 6 = 3x + 6 2y - 6 + 6 = 3x + 6 + 62y + 0 = 3x + 122y = 3x + 122y / 2 = (3x + 12) / 2y * (2 / 2) = (3x + 12) / 2y * (1) = (3x + 12) / 2y = (3x + 12) / 2y = (3/2)x + 6 Point Slope Form of Equation for Line “H”: y = (3/2)x + 6 ` The Standard Form `Standard FormAx + By = C A, B, and C are constantsy = (3/2)x + 6(3/2)x + 6 = y(3/2)x + 6 - 6 = y - 6 (3/2)x + 0 = y - 6 (3/2)x = y - 6 (3/2)x - y = y - 6 - y (3/2)x - y = y - 6 - y (3/2)x - y = y - y - 6(3/2)x - y = 0 - 6(3/2)x - y = -6 Standard Form of Equation for Line “H”: (3/2)x - y = -6 ` --------------------------------------------

 Dec 23, 2012 Parallel, Perpendicular, or Intersecting Lines by: Staff -------------------------------------------- Part V Are the two lines parallel, perpendicular, or intersecting, at a non-right angle? Explain your reasoning. Line “A” and Line “H” are: parallel, since they have the same slope Thanks for writing. Staff www.solving-math-problems.com