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Section 2.2 – Linear Equations in One Variable Linear Equations An equation that can be written in the form mx + b = 0, where m and b are real numbers is a linear equation. Solutions to a linear equation are values of x that make the statement true. Solve. Ex 1: Ex 2: Linear Equations Ex 3 : 2 6 1 5 x x 3 5 2 6 6 1 5 2 30 x x 30 5 2 6 3 20 x 36 15 25 x 5 x 36 15 5 x 51 51 x 5 Rational Equations We will solve rational equations by multiplying both sides of the equation by the LCD. The resulting equation will be a linear equation, which we know how to solve. The solutions that we obtain must be checked to determine if they are in the domain. Rational Equations Ex 3 : 1 1 3 x 5 2x LCD : 10x 1 1 3 10 x 10 x x 5 2x 10 2x 15 5 2x x 5 2 5 2 Slope The slope m of a line containing the points (x1, y1) and (x2, y2) is given by m rise run m m change in y change in x m y x y2 y1 y1 y2 x2 x1 x1 x2 ***The slope is a measure of the slant of a line. ***You are often asked to find the rate of change in/of y over x find the slope. Slope & Lines The slope of a line is constant. That is, the slope between every pair of points on a line is equal. y f(x)=2x-4 Series 1 5 rise m run 4 3 2 1 x -9 -8 -7 -6 -5 -4 -3 -2 -1 1 -1 -2 -3 -4 -5 2 3 4 5 6 7 8 9 Slope from Graph Find the slope of the line. Ex 5: Ex 6: m rise 3 run 5 x 2 m rise 3 Undefin ed run 0 The slope of a vertical line is undefined. Slope from Graph Find the slope of the line. y3 Ex 7: m rise 0 0 run 3 The slope of a h orizontal line is 0. Ex 8: m rise 5 run 8 Slope & Lines Decreasing Slope is negative Increasing Slope is positive Vertical Slope is undefined Horizontal Slope is 0 y 1 x4 Slope from Points Find the slope of the line containing the given points. Ex 9 : (9,8) (7,6) m 8 (6) 8 6 14 7 97 16 16 8 Ex 10 : ( 2 ,4) (0.56,4) m 4 (4) 44 0 0 2 0.56 2 0.56 2 0.56 Slope-Intercept Form The linear equation y = mx + b is written in slopeintercept form. y f(x) = mx + b The graph of an equation in this form is a straight line with a slope equal to m and a yintercept at the point (0, b). (0, b) x Graph Using Slope-Intercept Graph the equation using the slope and the y-intercept. Ex 11 : y 3 x 1 2 3 m 2 b 1 Slope from Graph Graph the linear equation and state its slope and y-intercept. Ex 12 : 2y x 8 2y x 8 1 y x4 2 1 f ( x) x 4 2 rise 1 run 2 b (0,4) m 9 8 7 6 y 5 4 3 2 1 -9 -8 -7 -6 -5 -4 -3 -2 -1 -1 -2 -3 -4 -5 -6 -7 -8 -9 x 1 2 3 4 5 6 7 8 9 Find an Equation for a Line Point-Slope Equation y y1 m( x x1 ) In order to find an equation of a line, you MUST have • slope m • point ( x1 , y1 ) Find an Equation for a Line Write a slope-intercept equation for a line with the given characteristics. 3 8 y - intercept (0,5) Ex 13 : m 3 y (5) ( x (0)) 8 3 y 5 ( x 0) 8 3 y 5 x 0 8 3 y x5 8 This line crosses the y axis at 5 and it decreases. 3 y x5 8 Find an Equation for a Line Ex 14 : m 2, passes through (5,1) m 2 (5,1) y (1) 2( x (5)) y 1 2( x 5) y 1 2 x 10 y 2 x 9 y 2 x 9 This line crosses the y axis at -9 and it decreases. Horizontal/Vertical Lines Write equations of the horizontal and vertical lines that pass through the given point. 1 Ex 15 : ,7 4 Horizontal: y7 Vertical: x Ex16 : 0.03,0 Horizontal: 1 4 Vertical: y0 x 0.03 Parallel/ Perpendicular Lines Parallel lines have slopes that are equal. m1 m2 Perpendicular lines have slopes that are negative reciprocals. m2 m 3 1 m1 m|| 3 m1 m2 1 1 m 3 Parallel/ Perpendicular Write a slope-intercept equation for a line passing through the given point that is parallel to the given line. Then write a second equation for a line passing through the point that is perpendicular to the given line. Ex 17 : (4,5), 2 x y 4 Parallel Line: m|| 2 y (5) 2( x (4)) y 5 2( x 4) y 5 2 x 8 y 2 x 13 y 2 x 13 m 2 (4,5) 2 x y 4 y 2 x 4 Perpendicular Line: m y (5) 1 ( x (4)) 2 1 ( x 4) 2 1 y5 x2 2 1 y x 3 2 1 2 y5 1 y x3 2 Parallel/ Perpendicular 1 y x3 2 Graph all lines on the same set of axes. y 2 x 4 y 2 x 13 y f(x)=-2x - 4 f(x)=-2x - 13 f(x)=(1/2)x - 3 1 x -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 1 -1 -2 -3 -4 -5 -6 -7 -8 2 3 4 5 6 7 8 Parallel/ Perpendicular Ex 18 : (4,5), y 1 m 0 (4, 5) y f(x)=-1 undefined f(x)=-5 m|| 0 Parallel Line: y 5 Perpendicular Line: m 9 8 x=4 x4 7 6 5 4 3 2 1 x -9 -8 -7 -6 -5 -4 -3 -2 -1 1 -1 -2 -3 -4 -5 -6 -7 -8 -9 2 3 4 5 6 7 8 9