Also…get a sheet of graph paper! 5.3: Writing Linear Equations Given Two Points Homework 43: p.288: 21-35, 45-47 All Learning Objectives: Write an equation of a line given two point on the line Consider the equation 3𝑥−4𝑦=8. Solve the equation for 𝑦. What do you notice about the coefficients of the original equation with the slope of the new equation? The slope is the opposite sign of the first coefficient over the second coefficient Also…get a sheet of graph paper!

Concept Linear equations written in slope-intercept form require a slope and an intercept. If the slope is unknown, one must be found. Two points are required to find the slope.

Example 1: Writing a Linear Equation Given Two Points Write a linear equation given the points 1, 6 , (3,−4) Step 1: Determine a Slope 𝑚= −4−6 3−1 = −10 2 =−5

Example 1: Writing a Linear Equation Given Two Points Write a linear equation given the points 1, 6 , (3,−4) Step 2: Use the slope and one of the two points, find b 𝑦=𝑚𝑥+𝑏 6=−5(1)+𝑏 −4=−5(3)+𝑏 6=−5+𝑏 −4=−15+𝑏 +𝟓 +𝟓 +𝟏𝟓 +𝟏𝟓 11=𝑏

Student Led Example 1: Writing Equations Given Two Points Write a linear equation given two points −3, 1 , (6, 7) 𝒚= 𝟐 𝟑 𝒙+𝟑 A 8, 9 , (−4, 0) 𝒚= 𝟑 𝟒 𝒙+𝟑 B −4, 3 , (0, −1) C 𝒚=−𝒙−𝟏

Activity: Slopes of Perpendicular Lines Get a protractor, graph paper and a straightedge. Draw an xy-coordinate plane Graph the equation: 𝑦=− 3 4 𝑥+1 Use the protractor to draw a perpendicular line (90°) so that it intersects at the 𝑦− intercept Find the slope (rise over run) of the perpendicular line

Concept: Slopes of Perpendicular Line If 𝑚 is the slope of a line, then 𝑚 ⊥ is the opposite reciprocal of 𝑚 Opposite = changing signs Reciprocal = upside down fraction 𝑚⋅ 𝑚 ⊥ =−1

Concept: Perpendicular Lines

Example 2: Identifying Perpendicular Lines Identify which lines are perpendicular: 𝑦=3𝑥−2, 𝑦=− 1 4 𝑥+2 𝑦=4𝑥 𝑦=− 1 3 𝑥−7

Student Led Example 2: Identifying Perpendicular Lines Identify which lines are perpendicular: 𝑦=2𝑥 𝑦=−2𝑥−7 𝑦=7𝑥+1 𝑦=− 1 7 𝑥−21

Example 3: Write an Equation of a Perpendicular Line Given the points −6, −11 , 3, −5 , write a linear equation of a perpendicular line which passes through (2, −1)

Example 3: Write an Equation of a Perpendicular Line Given the points −6, −11 , 3, −5 , write a linear equation of a perpendicular line which passes through (2, −1) Step 1: Find the slope 𝑚= −5− −11 3− −6 = −5+11 3+6 = 6 9 = 2 3 Step 2: Find the Perpendicular Slope 𝑚 ⊥ =− 3 2

Example 3: Write an Equation of a Perpendicular Line Given the points −6, −11 , 3, −5 , write a linear equation of a perpendicular line which passes through (2, −1) Step 3: Write the Equation →−1=− 3 2 (2)+𝑏 𝑦=𝑚𝑥+𝑏 −1=−3+𝑏 →2=𝑏 𝑦=− 3 2 𝑥+2

Example 3: Write an Equation of a Perpendicular Line Given the points −6, −11 , 3, −5 , write a linear equation of a perpendicular line which passes through (2, −1) Step 3: Check by Graphing 𝟐, −𝟏 𝟑, −𝟓 (−𝟔,−𝟏𝟏)

Student Led Example 3: Perpendicular Lines Write an equation in slope-intercept form for the line that passes through (–5, 3) and is perpendicular to the line described by 𝑦 = 5𝑥. 𝑦=− 1 5 𝑥+2

Example 4: Geometric Application Show that ABC is a Right Triangle Slope of 𝐴𝐵 𝑚= 2−0 1+2 = 2 3 Slope of 𝐵𝐶 𝑚= 2+3 1−0 = 5 1 Slope of 𝐴𝐶 𝑚= −3−0 0+2 =− 3 2

Student Led Example 4: Geometric Application Show that PQR is a Right Triangle

Exit Task: Given the equation 𝑦=− 1 2 𝑥−6 End of Lesson Exit Task: Given the equation 𝑦=− 1 2 𝑥−6 Write an equation parallel passing through the point (8,5) Write an equation perpendicular pass through the point (8,15) 𝒚 𝒑𝒂𝒓𝒂 =− 𝟏 𝟐 𝒙+𝟗 𝒚 𝒑𝒆𝒓𝒑 =𝟐𝒙−𝟏