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Slopes of Lines Slope = rise = run Lines that have a positive slope

Slopes of Lines Slope = rise = run Lines that have a positive slope ‘ rise ’ through the x-y plane. The greater the slope of a line, the steeper the line will be. ( smallest slope – blue line biggest slope – green line). Lines with negative slope

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Slopes of Lines Slope = rise = run Lines that have a positive slope

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  1. Slopes of Lines Slope = rise = run Lines that have a positive slope ‘rise’ through the x-y plane. The greater the slope of a line, the steeper the line will be. ( smallest slope – blue line biggest slope – green line)

  2. Lines with negative slope ‘fall’ through the x-y plane The greater the ‘absolute value’ Of the slope the steeper the ‘fall’ will be. Slope-Intercept form:y = mx + b m = slope; b = y-intercept Therefore to quickly find the slope and y-intercept, ‘Dah’—Just solve the equation for y!

  3. This information allows us to graph lines quickly and easily. Example 1: Find the slope and y-intercept of 7x + 13y = 26 1st: Isolate the ‘y’ term: -7x -7x 13y = -7x + 26 2nd: Get ‘y’ all alone 13 13 Voila  y = (-7/13)x + 2 Align y = m x + b To see that m = -7/13 and that b = 2 Now Graph!

  4. All non vertical lines are parallel - (ll) - if they have the • ‘same slope’! • Slopes of perpendicular lines – ‘ l .’ – are • ‘opposite reciprocal slopes’ • Example 2: A line, ‘l’ has equation x + 2y = 5 • What is the slope of the line: ?? • (solve for y to find the ‘m’ value) • Parallel to ‘l’b) Perpendicular to ‘l’ • (To be (ll) the line must have • the same ‘m’ value) (Here the ‘m’ value must be the • opposite reciprocal) • m = -1/2 m = 2/1 or 2

  5. The graphs below illustrate the effects of ‘m’ and ‘b’ on an equation written in slope-intercept form. Guess what is common about the lines in the two graphs. What is common What is common here? here? same y-intercept same slope

  6. Remember: Two lines are parallel if they have ??? • And two lines are perpendicular if they have ??? • Example 3: The equations of three lines are given. • Which lines are parallel (ll) and which lines are • perpendicular ( l .)? • (Solve each equation for y – then compare the ‘m’ vlaues) • y = (3/4)x – 7 b) 4x + 3y = 10 c) 6x – 8y = 11 • Example 4: Find the value of ‘k’ if the line joining • (2,k) with (4,5) is:

  7. Parallel to y = 3x+1 • (Using the points (2,k) and (4,5) ‘Calculate the slope: ?? • k – 5 • 2 – 4 • k – 5 Now set this equal to 3 • - 2 1 • proportion: k – 5 = 3 • - 2 1 • cross multiply and solve for k. • k – 5 = - 6 • + 5 + 5 • k = - 1 • Perpendicular to y = 3x+1 • now set k – 5 = - 1 (opposite reciprocal slope) • - 2 3 • Cross multiply and solve for ‘k’ • 3k – 15 = 2 • 3k = 17 • k = 17 • 3

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