1.4: equations of lines CCSS: G.GPE.5 Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point). GSE M(G&M)–10–9 Solves problems on and off the coordinate plane involving distance, midpoint, perpendicular and parallel lines, or slope.

Various Forms of an Equation of a Line. Slope-Intercept Form Point-Slope Form

EXAMPLE 1 Write an equation given the slope and y-intercept Write an equation of the line shown.

Write an equation given the slope and y-intercept EXAMPLE 1 Write an equation given the slope and y-intercept SOLUTION From the graph, you can see that the slope is m = and the y-intercept is b = –2. Use slope-intercept form to write an equation of the line. 3 4 y = mx + b Use slope-intercept form. y = x + (–2) 3 4 Substitute for m and –2 for b. 3 4 3 4 y = x (–2) Simplify.

GUIDED PRACTICE Write an equation of the line that has the given slope and y-intercept. 1. m = 3, b = 1 3. m = – , b = 3 4 7 2 ANSWER ANSWER y = – x + 3 4 7 2 y = x + 1 3 2. m = –2 , b = –4 ANSWER y = –2x – 4

Write an equation given the slope and a point EXAMPLE 2 Write an equation given the slope and a point Write an equation of the line that passes through (5, 4) and has a slope of –3. SOLUTION Because you know the slope and a point on the line, use point-slope form to write an equation of the line. Let (x1, y1) = (5, 4) and m = –3. y – y1 = m(x – x1) Use point-slope form. y – 4 = –3(x – 5) Substitute for m, x1, and y1. y – 4 = –3x + 15 Distributive property y = –3x + 19 Write in slope-intercept form.

Slope Formula y2 – y1 x2 – x1 The slope of a line through the points (x1, y1) and (x2, y2) is as follows: y2 – y1 x2 – x1 m =

Example 1 Write the equation of the line with slope = -2 and passing through the point (3, -5). Substitute m and into the Point-Slope Formula.

Example 2: Point Slope Form Y X Let’s find the equation for the line passing through the points (3,-2) and (6,10) First, Calculate m : Y-axis X-axis (6,10) = (10 – -2) DY 12 = m = = 4 DX (6 – 3) 3 DY (3,-2) DX

Example 2: Point Slope Form Y X To find the equation for the line passing through the points (3,-2) and (6,10) Next plug it into Point Slope From : y – y1 = m(x – x1) Select one point as P1 : Y-axis X-axis (6,10) Let’s use (3,-2) The Equation becomes: y – -2 = 4(x – 3) DY (3,-2) DX

Example 2: Point Slope Form Y X If you want ….simplify the equation to put it into Slope Intercept Form Distribute on the right side and the equation becomes: y + 2 = 4x – 12 Subtract 2 from both sides gives. Y-axis X-axis y + 2 = 4x – 12 -2 = - 2 (6,10) y = 4x – 14 DY (3,-2) DX

Write an equation given two points Ex3 EXAMPLE 4 Write an equation given two points Write an equation of the line that passes through (5, –2) and (2, 10). SOLUTION The line passes through (x1, y1) = (5,–2) and (x2, y2) = (2, 10). Find its slope. y2 – y1 m = x2 – x1 10 – (–2) = 2 – 5 12 –3 = –4

Write an equation given two points EXAMPLE 4 Write an equation given two points You know the slope and a point on the line, so use point-slope form with either given point to write an equation of the line. Choose (x1, y1) = (4, – 7). y2 – y1 = m(x – x1) Use point-slope form. y – 10 = – 4(x – 2) Substitute for m, x1, and y1. y – 10 = – 4x + 8 Distributive property y = – 4x + 8 Write in slope-intercept form.

Example 3 Write the equation of the line that goes through the points (3, 2) and (5, 4).

Understanding Slope Two (non-vertical) lines are parallel if and only if they have the same slope. (All vertical lines are parallel.) Parallel lines have the __________ slope

Understanding Slope The slope of AB is: The slope of CD is: Since m1=m2, AB || CD

Perpendicular Lines (┴)Perpendicular Lines- 2 lines that intersect forming 4 right angles Right angle

Slopes of  Lines 2 lines are  their slopes are opposite reciprocals of each other; such as ½ and -2. Vertical and horizontal lines are  to each other.

Example Line l passes through (0,3) and (3,1). Line m passes through (0,3) and (-4,-3). Are they ? Slope of line l = Slope of line m = l  m Opposite Reciprocals!

4 -1 2 y = - x 1 Let's look at a line and a point not on the line Let's find the equation of a line parallel to y = - x that passes through the point (2, 4) y = - x What is the slope of the first line, y = - x ? (2, 4) 1 This is in slope intercept form so y = mx + b which means the slope is –1. So we know the slope is –1 and it passes through (2, 4). Having the point and the slope, we can use the point-slope formula to find the equation of the line 4 -1 2 Distribute and then solve for y to leave in slope-intercept form.

What if we wanted perpendicular instead of parallel? Let's find the equation of a line perpendicular to y = - x that passes through the point (2, 4) y = - x (2, 4) The slope of the first line is still –1. The slope of a line perpendicular is the negative reciporical so take –1 and "flip" it over and make it negative. 4 1 2 Distribute and then solve for y to leave in slope-intercept form. So the slope of a perpendicular line is 1 and it passes through (2, 4).

Slope-Intercept Form (y = mx+b) Find the equation of a line passing through the points P(0, 2) and Q(3, –2). Is this line parallel to a line with the equation

a) Find the equation of a line that passes through the point G ( -4, 5) and is perpendicular to b) Write the equation of a line that passes through point P (1, -2) and is parallel the line that passes through points A (-4,6) and B ( 4,10)

What is the equation of a line that models a line that is perpendicular to and goes through the point (6,6) ?

Graphing Equations Conclusions What are the similarities you see in the equations for Parallel lines? What are the similarities you see in the equations for Perpendicular lines? What are the differences between parallel and perpendicular lines? What similarities are there in equations of lines written in slope intercept and point slope form? What difference are there in equations of lines written in slope intercept and point slope form?

Equation Summary Slope: Slope-Intercept Form: Point-Slope Form: Vertical change (DY) Slope (m) = Horizontal change (DX) Slope-Intercept Form: y = mx + b Point-Slope Form: y – y1 = m(x – x1)