warm-up 1) find the slope of the line containing the points (-1,12) and (5,-6). 4 minutes 2)...
TRANSCRIPT
Warm-Up1) Find the slope of the line containing the points (-1,12) and (5,-6).
4 minutes
2) Identify the slope and y-intercept for the line y = -5x + 7.
3) Write the equation in slope-intercept form for the line that has a y-intercept of 0 and a slope of -1.
1.3.1 Linear Equations in Two 1.3.1 Linear Equations in Two VariablesVariables
1.3.1 Linear Equations in Two 1.3.1 Linear Equations in Two VariablesVariables
Objectives: •Write a linear equation in two variables given sufficient information
Example 1Write an equation for the line containing the points (1,5) and (2,8).
change in y-coordinatesm
change in x-coordinates
2 1
2 1
y ym
x x
8 5m
2 1
3m 3
1
Example 1Write an equation for the line containing the points (1,5) and (2,8).
m 3
y = mx + b5 =3(1)+
b5 = 3 + b-3 -32 = b
y = 3x + 2
substitute
simplify
The Point-Slope Equation
y – y1 = m(x – x1)
Example 2Write an equation for the line with slope 7 that contains the point (3,4).
y – y1 = m(x – x1)y – 4 = 7(x – 3)
y – 4 = 7x - 21+4
+4y = 7x - 17
Example 3Write an equation for the line containing (5,7) and (2,1).
y – y1 = m(x – x1)
First, find the slope: 2 1
2 1
y ym
x x
1 7m
2 5
63
2
y – 7 = 2(x – 5)
y – 7 = 2x - 10+7
+7y = 2x - 3
Example 4Write an equation for the line shown below.
-4 -2
2
42
4
-4
-2
First, find any two points on the line.
(-3,-3) and
(1,-1)
2m
4
12
y – y1 = m(x – x1) y + 3 = ½(x +
3)y + 3 = ½x + 1½ -3 -3
y = ½x – 1½
Example 5Marva left her house and drove at a constant speed to a conference in another state. She picked up Delia along the way. Two hours after picking up Delia, they were 140 miles from Marva’s house, and 5 hours after picking up Delia, they were 344 miles from Marva’s house. How far from her house was Marva when she picked up Delia?
Homework
p.26 #11-31 odds
Warm-UpWrite an equation for the line given the indicated points.
4 minutes
1) (-6,-6) and (-3,1)
Write an equation in point-slope form for the line that has the indicated slope, m, and contains the given point.
2) m = 4; (9,-3)
1.3.2 Linear Equations in Two 1.3.2 Linear Equations in Two VariablesVariables
1.3.2 Linear Equations in Two 1.3.2 Linear Equations in Two VariablesVariables
Objectives: •Write an equation for a line that is parallel or perpendicular to a given line
ActivityGraph y = 2x + 1.
On the same screen, graph y = 2x, y = 2x – 2.5, and y = 3x + 1.
Which equations have graphs that appear to be parallel to that of y = 2x + 1?y = 2x, y = 2x –
2.5What do these equations have in common?same slopes
Write an equation in slope-intercept form for a line whose graph you think will be parallel to that of y = 2x + 1. Verify by graphing.
ActivityGraph y = 2x + 1.
On the same screen, graph
1y x 2,
2
1y x 2,
2 1
and y x 32
Which equations have graphs that appear to be perpendicular to that of y = 2x + 1?
1y x 2
2 1
and y x 32
What do these equations have in common?
their slopes are negative reciprocals of each other
Activity
Write an equation in slope-intercept form whose graph you think will be perpendicular to that of y = 2x + 1. Verify by graphing.
Parallel LinesParallel lines are lines in the same plane that never intersect.
-8 -6 -4 -2
2
42 6 8
4
6
-4
-6
-8
-2
8
Parallel lines have the same slope.
Perpendicular LinesPerpendicular lines are lines that intersect to form a 900 angle.
-8 -6 -4 -2
2
42 6 8
4
6
-4
-6
-8
-2
8
The product of the slopes of perpendicular lines is -1.
4m 2
2
2 1m
4 2
Example 1Determine whether these lines are parallel or perpendicular.
y – 2 = 5x + 4
and -15x + 3y = 9+2 +2
y = 5x + 6
+15x +15x 3y = 9 +
15x3 3
y = 3 + 5x
y = 5x + 3
The lines have the same slope.So they are parallel.
Example 2Write an equation in slope-intercept form for the line containing (-3,-5) and parallel to the line y = 2x + 1.First, we need the slope of the line y = 2x + 1.m = 2
Second, we need to find out the slope of the line that is parallel to y = 2x + 1.m 2
Lastly, we use the point-slope formula to find our equation.
1 1(y y ) m(x x ) (y 5) 2(x 3)
y 5 2(x 3) y + 5 = 2x + 6 y = 2x + 1
Example 3Write an equation for the line containing (-3,-5) and perpendicular to the line y = 2x + 1.First, we need the slope of the line y = 2x + 1.
m = 2Second, we need to find out the slope of the line that is perpendicular to y = 2x + 1. 1
m2
Lastly, we use the point-slope formula to find our equation.
1 1(y y ) m(x x ) 1
(y 5) (x 3)2
1y 5 (x 3)
2
1 3y 5 x
2 2
1 13y x
2 2
Homework
p.26 #35-51 odds