2. Linear Equations Equations of the form ax + by = c are called linear equations in two variables . The point (0,4) is the y -intercept . The point (6,0) is the x -intercept . x y 2 -2 This is the graph of the equation 2 x + 3 y = 12. (0,4) (6,0)
3. Slope of a Line The slope of a line is a number, m , which measures its steepness. m = 0 m = 2 m is undefined y x 2 -2 m = 1 2 m = - 1 4
4. Slope Formula The slope of the line passing through the two points ( x 1 , y 1 ) and ( x 2 , y 2 ) is given by the formula The slope is the change in y divided by the change in x as we move along the line from ( x 1 , y 1 ) to ( x 2 , y 2 ). x y x 2 – x 1 y 2 – y 1 change in y change in x y 2 – y 1 x 2 – x 1 m = , ( x 1 ≠ x 2 ). ( x 1 , y 1 ) ( x 2 , y 2 )
5. Example: Find Slope Example : Find the slope of the line passing through the points (2 , 3) and (4, 5). Use the slope formula with x 1 = 2 , y 1 = 3 , x 2 = 4 , and y 2 = 5 . 2 2 (2, 3) (4, 5) y 2 – y 1 x 2 – x 1 m = 5 – 3 4 – 2 = = 2 2 = 1 x y
6. Slope-Intercept Form A linear equation written in the form y = mx + b is in slope-intercept form . To graph an equation in slope-intercept form : 1. Write the equation in the form y = mx + b. Identify m and b . The slope is m and the y-intercept is (0, b ). 2. Plot the y -intercept (0, b ). 3. Starting at the y -intercept, find another point on the line using the slope. 4. Draw the line through (0, b ) and the point located using the slope.
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8. Point-Slope Form A linear equation written in the form y – y 1 = m ( x – x 1 ) is in point-slope form . The graph of this equation is a line with slope m passing through the point ( x 1 , y 1 ) . Example : The graph of the equation y – 3 = - ( x – 4) is a line of slope m = - passing through the point (4, 3). 1 2 1 2 (4, 3) m = - 1 2 x y 4 4 8 8
9. Example: Point-Slope Form Example : Write the slope-intercept form for the equation of the line through the point (-2, 5) with a slope of 3. Use the point-slope form, y – y 1 = m ( x – x 1 ) , with m = 3 and ( x 1 , y 1 ) = (-2, 5). y – y 1 = m ( x – x 1 ) Point-slope form y – y 1 = 3 ( x – x 1 ) Let m = 3. y – 5 = 3 ( x – ( -2 )) Let ( x 1 , y 1 ) = (-2, 5) . y – 5 = 3( x + 2) Simplify. y = 3 x + 11 Slope-intercept form
10. Example: Slope-Intercept Form Example : Write the slope-intercept form for the equation of the line through the points (4, 3) and (-2, 5). y – y 1 = m ( x – x 1 ) Point-slope form Slope-intercept form y = - x + 13 3 1 3 2 1 5 – 3 -2 – 4 = - 6 = - 3 Calculate the slope. m = Use m = - and the point (4, 3). y – 3 = - ( x – 4 ) 1 3
11. Example: Parallel Lines Two lines are parallel if they have the same slope. If the lines have slopes m 1 and m 2 , then the lines are parallel whenever m 1 = m 2 . Example : The lines y = 2 x – 3 and y = 2 x + 4 have slopes m 1 = 2 and m 2 = 2. The lines are parallel. x y y = 2 x + 4 (0, 4) y = 2 x – 3 (0, -3)
12. Example: Perpendicular Lines Two lines are perpendicular if their slopes are negative reciprocals of each other. If two lines have slopes m 1 and m 2 , then the lines are perpendicular whenever The lines are perpendicular. or m 1 m 2 = -1. y = 3 x – 1 (0, 4) (0, -1) 1 m 1 m 2 = - x y y = - x + 4 1 3 Example : The lines y = 3 x – 1 and y = - x + 4 have slopes m 1 = 3 and m 2 = - . 1 3 1 3
