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3-6C Lines in the Coordinate Plane Objectives: G.GPE.5: Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems. For the Board: You will be able to graph lines and write their equations in slope-intercept and point-slope form. You will be able to classify lines as parallel, intersecting, or coinciding. A system of two linear equations in two variables represents two lines. The lines can be parallel, intersecting, or coinciding. Perpendicular lines are a special form of intersecting lines. Coinciding lines are the same line but written in a different form. Pairs of Lines Parallel Lines Same slope Different y-intercept y = 5x + 8 y = 5x – 4 Intersecting Lines Different slopes y = 2x – 5 y = 4x + 3 Perpendicular Lines Slopes which are opposites and reciprocals y = 2x – 5 y = -1/2 x + 3 Coinciding Lines Same slope Same y-intercept y = 2x – 4 y = 2x - 4 Steps: 1. Rearrange both equations into slope-intercept form by solving the equation for y. 2. Compare the slopes using the above table. 3. If the slopes are equal, compare the y-intercepts. Open the book to page 192 and read example 3. Example: Determine whether the lines 3x+ 5y = 2 and 3x + 6 = -5y are parallel, intersecting, perpendicular or coinciding. 3x + 5y = 2: 5y = -3x + 2 y = -3/5 x + 2/5 3x + 6 = -5y: 5y = -3x – 6 y = -3/5 x – 6/5 Slope are the same but y-intercepts are different so the lines are parallel. White Board Activity: Practice 3: Determine whether the lines are parallel, intersecting, perpendicular or coinciding. a. y = 3x + 7, y = -3x – 4 intersecting: they have different slopes b. y = -1/3 x + 5, 6y = -2x + 12 y = -2x/6 + 12/6 or y = -1/3 x + 2 parallel: they have the same slope and different y-intercepts c. 2y – 4x = 16, y – 10 = 2(x – 1) 2y = 4x + 16 or y = 4/2 x + 16/2 or y = 2x + 8 y – 10 = 2x – 2 or y = 2x – 2 + 10 or y = 2x + 8 coinciding: they have the same slope and the same y-intercept Write the equation of a line parallel to a given equation through a given point. Steps: 1. Solve the given equation for y. 2. Determine its slope. 3. Since parallel lines have the same slope, use this slope for the new equation. 4. Use the point-slope formula. Use the determined slope and the given point. 5. Solve the equation for y to get slope-intercept form if required. Example: Write the equation of the line parallel to 2x – 3y = 6 through (4, 5). 2x – 3y = 6: -3y = -2x + 6 y = -2/-3 x + 6/-3 y = 2/3 x – 2 Slope = 2/3 y – 5 = 2/3 (x – 4) y – 5 = 2/3 x – 8/3 y = 2/3 x – 8/3 + 5 y = 2/3 x – 8/3 + 15/3 y = 2/3 x + 7/3 White Board Activity Practice: Write the equation of a line parallel to a given equation through a given point. a. y = 3x – 7, (1, 8) Since the equation is already solved for y. The slope is the coefficient of the x term. Slope = 3 y – 8 = 3(x – 1) y – 8 = 3x – 3 y = 3x – 3 + 8 y = 3x + 5 b. 3x + 4y = 12, (4, -3) First solve the equation for y. 4y = 12 – 3x y = 12/4 – 3x/4 y=3–¾x The slope is the coefficient of the x term. Slope = -3/4 y + 3 = -3/4 (x – 4) y + 3 = -3/4 x + (-3/4)(-4/1) y + 3 = -3/4 x + 12/4 y + 3 = -3/4 x + 3 y = -3/4 x + 3 – 3 y = -3/4 x Write the equation of the line perpendicular to the given line through the given point. Steps: 1. Solve the given equation for y. 2. Determine its slope. 3. Perpendicular lines have the opposite-reciprocal slopes. Determine the slope for the new equation. 4. Use the point-slope formula. Use the determined slope and the given point. 5. Solve the equation for y to get slope-intercept form if required. Example: Write the equation of the line parallel to 2x – 3y = 6 through (4, 5). From above the slope is 2/3. Use -3/2 in the equation. y – 5 = -3/2 (x – 4) y – 5 = -3/2 x + 12/2 y – 5 = -3/2 x + 6 y = -3/2 x + 6 + 5 y = -3/2 x + 11 White Board Activity: Practice: Write the equation of the line perpendicular to the given line through the given point. a. y = -3/2 x + 7, (6, 0) Since the equation is already solved for y. The slope is the coefficient of the x term. Slope = -3/2 so use its opposite-reciprocal. Slope = 2/3. y – 0 = 2/3(x – 6) y = 2/3 x + (2/3)(-6/1) y = 2/3 x + (-12/3) y = 2/3 x – 4 b. 4x – 2y = 8, (-5, 4) Solve the equation for y. -2y = 8 – 4x y = 8/-2 – 4x/-2 y = -4 + 2x Slope = 2 so use its opposite-reciprocal. Slope = -1/2. y – 4 = -½(x + 5) y – 4 = -1/2x + (-1/2)(5/1) y – 4 = -1/2 x – 5/2 y = -1/2 x – 5/2 + 4/1 y = -1/2 x – 5/2 + 8/2 y = -1/2 x + 3/2 Assessment: Question student pairs. Independent Practice: Text: pg. 194 – 195 prob. 8 – 11, 19 – 22, 33 – 40. For a Grade: Text: pgs. 194 – 195 prob. 22, 36, 38.