Unit 1 Introduction to Precalculus Linear Equations in Two Variables (Unit 1.3)

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1 Unit 1 Introduction to Precalculus Linear Equations in Two Variables (Unit 1.3) William (Bill) Finch Mathematics Department Denton High School Lesson Goals When ou have completed this lesson ou will: Find the slope of a line and understand it as the a rate of change between two variables. Use slope to graph linear equations in 2 variables. Use slope to identif parallel and perpendicular lines. Write linear equations in 2 variables, including Point-slope form and Slope-intercept form. Write linear equations in 2 variables, including Point-slope form and Slope-intercept form. Lin. Eq. 2 /19

2 Lesson Goals When ou have completed this lesson ou will: Use slope and linear equations in two variables to model and solve real-life problems. Lin. Eq. 3 /19 Slope as a Rate of Change The simplest mathematical model is the Linear Equation = m + b where b is the -intercept (a point) and m is the slope (a rate of change). (2, 2) (1, 1) m units b 1 unit m = m = slope = rate of change between variables Lin. Eq. 4 /19

3 ( ) Four Possible Slopes of a Line m = (a) Positive slope (b) Negative slope (c) Zero slope (d) Undefined slope Lin. Eq. 5 /19 Eample 1 Find the slope of a line passing through ( 3, 4) and (1, 6). Lin. Eq. 6 /19

4 Eample 2 Determine the slope of the line shown below Lin. Eq. 7 /19 Equations of Lines General form: A + B + C = 0 Vertical line: Horizontal line: Slope-intercept form: = a = b = m + b Point-slope form: 1 = m( 1 ) Point-point form: 1 = 2 1 ( 1 ) 2 1 Intercept form: a + = 1, -int = a, -int = b b Lin. Eq. 8 /19

5 Writing Linear Equations To write the equation of a line ou need two pieces of information: 1. a point 2. the slope The point is a location and the slope describes how to find other points along the line. Note the point does not have to be the -intercept! Lin. Eq. 9 /19 Writing Linear Equations Choose the form in which to write an equation based on given information: Given a point (not the -intercept) and a slope Point-slope form Given a point (the -intercept) and a slope Slope-intercept form Even if ou want the final answer to be in Slope-intercept form, if all ou have is some point and a slope then ou can write the equation in Point-slope form and solve the resulting equation for. Lin. Eq. 10 /19

6 Eample 3 Write the equation of a line passing through the point whose coordinates are ( 2, 3) and with a slope of 4. Write our 3 final answer in slope-intercept form. Lin. Eq. 11 /19 Eample 4 Write the equation of a line passing through the points (1, 4) and (5, 2). Lin. Eq. 12 /19

7 Eample 5 Write the equation of a line passing through (0, 3) with a slope of 2. Lin. Eq. 13 /19 Parallel and Perpendicular Lines A pair of parallel or perpendicular lines each have a special relationship between their slopes: Parallel lines have slopes equal to each other (m 1 = m 2 ). Perpendicular lines have slopes that are negative-reciprocals of each other (m 1 m 2 = 1). Lin. Eq. 14 /19

8 Eample 6 A line passes through the point (1, 3). Find the equation of a line: a. Parallel to the line whose equation is 3 = 2( + 1). b. Perpendicular to the line whose equation is = 0. Lin. Eq. 15 /19 Eample 7 Write the equation of the perpendicular bisector of the segment joining A( 2, 3) and B(4, 5). Lin. Eq. 16 /19

9 Graphing Calculator Tip If ou want to use a graphing calculator to check our answers to the previous eamples of parallel or perpendicular lines, note that the slopes of lines will appear to be distorted unless ou select the ZSquare option in the Zoom menu. Lin. Eq. 17 /19 Eample 8 The sales per share for Starbucks Corporation were $6.97 in 2001 and $8.47 in Using onl this information, write a linear equation that gives the sales per share in terms of the ear. Let represent the ear with (with 2000 being set to 0) and the price of a share. Then predict the sales per share for Lin. Eq. 18 /19

10 What You Learned You can now: Find the slope of a line and understand it as the a rate of change between two variables. Use slope to graph linear equations in 2 variables. Use slope to identif parallel and perpendicular lines. Write linear equations in 2 variables, including Point-slope form and Slope-intercept form. Use slope and linear equations in two variables to model and solve real-life problems. Lin. Eq. 19 /19

LINEAR EQUATIONS IN TWO VARIABLES

LINEAR EQUATIONS IN TWO VARIABLES What You Should Learn Use slope to graph linear equations in two " variables. Find the slope of a line given two points on the line. Write linear equations in two variables.