1. Lines in the plane
Presented by group 4

2. Slope of a line
The slope of the non vertical line through the points (x1,y1) and (x2,y2) is
y = y2-y1
m= ___ ______
x x2-x1

3. Slope of a line Rules of slope:
If it is vertical, then x1 = x2 and the slope is undefined.
If it is horizontal, then y1 = y2 and the slope is zero.

4. Finding the slope of a line Example 1 & 2
Find the slope of the line through the two points. Sketch a graph of the line.
A) (-1,2) and (4,-2)
B) (1,1) and (3,4)

5. Example answers! A) y2 - y1 (-2) – 2 4
x2 - x – (-1)
B) y2 - y –
m = ________ = _________ = _____
x2 - x

6. Homework Questions!
Find the slope of the line through the pair of points.
1. (-3,-5) and (4,9)
2. (5,-3) and (-4,12)

7. Point-slope form equation of a line
What you need to know: coordinates of one point, slope.
y – y1
m = __________
x – x1
Point-slope form of an equation of a line passes through the point (x1,y1) and has slope m is
y – y1 = m(x – x1)

8. Example 1
Use the point-slope form to find an equation of the line that passes through the point (-3,-4) and has slope 2.
What to do?
Substitute x1 = -3 , y1 = -4 , and m = 2 into the point slope form and simplify.
y – y1 = m(x – x1) point-slope form
y – (-4) = 2(x – (-3)) substitute
y + 4 = 2x – 2(-3) distribute
y + 4 = 2x + 6
y = 2x + 2
[resulting in the slope intercept form of y = mx + b]

9. Homework Questions!
Use the point-slope form to find an equation of the line that passes through the points of
1. (-2,3) and (4,y) slope -3
2. (-3,-5) and (4,y) slope 3

10. Slope-intercept form equation of a line
What you need to know: y intercept, slope.
Y intercept (of a non vertical line): is the point where the line intersects the y-axis.
Slope-intercept form of an equation of a line: with slope m and y-intercept (0,b) is
y = mx + b

11. Example 1
Write the equation of the line with slope 3 passes through the point (1,-6) using the slope-intercept form.
What to do?
y = mx + b slope-intercept form
y = 3x + b m = 3
6 = 3(-1) + b y = 6 when x = -1
b = 9

12. Continued B B General form: Ax + By + C = 0
-where A and B are both not zero.
IF B is not equal to zero, the general form can be changed to the slope intercept form as follows.
Ax + By + C = 0
By = -Ax – C
Y = - A C
____ X + _ _____
B B
(Slope^) (y intercept^)

13. Homework Questions! Find the point slope form equation 1.(1,4) slope 2
Find a general form equation
1. (-7,-2) and (1,6)

14. Graphing Linear Equations in two variables
The standard form of a linear equation is Ax + By = C
If B = 0, the line is vertical
If A = 0, the line is horizontal

15. To graph: Rewrite in the form of y= Enter into y= Graph
*May need to adjust window*
Example:
Standard Form: 2x + 3y = 6
Solve for y: 3y = 6 – 2x
y = -⅔x + 2

16. Parallel and Perpendicular Lines
Rules:
Two non-vertical lines are parallel if and only if their slopes are equal.
Two non-vertical lines are perpendicular if and only if their slopes are opposite reciprocals. -1
m1 = _____ m2

17. Continued Find the point-slope form equation for the line.
y-y1=m(x-x1) Point-Slope formula
y-(-3)=1/4(x-2) Substitute
y+3=1/4x-2/4 Distributive Property
y=1/4x-7/2
Graph these two lines in your calculator and you will see that they appear perpendicular.

18. Finding a parallel line Example 1:
Find the equation of the line through P(1,-2) that is parallel to the line L with equation 3x-2y=1.
Convert the line to slope intercept form to find its slope.
3x-2y= Equation
-2y= -3x Subtract 3x
y=(3/2x)-(1/2) Divide by -2.
The slope of L is 3/2.
The line we're trying to find contains the point (1,-2). Use the point-slope form equation for the line you're trying to find.
y-y1 = m(x-x1) Equation
y-(-2) = 3/2x(x-1) Substitute
y+2 = 3/2x-3/ Distributive Property
y=3/2x - 7/2

19. Finding the equation of a perpendicular line
Find an equation of the line through P(2,-3) that is perpendicular to the line L with equation 4x+y=3.
-Convert the line to slope intercept form.
4x+y= Equation for L
y=-4x Subtract 4x.
The slope of L is -4.
The line whose equation we seek has slope 1/4 and passes through the point (2,-3).

20. Homework question!
Find an equation for the line y=3x-2 passing through point (1,2). Then find an equation for the line perpendicular to the line above.

21. Applying Linear Equations in Two Variables
You may use linear equations and their graphs frequently in application problems!

22. Example: Depreciation of real estate
Apartments purchased at $75,000 depreciates $2500 per year over a 30 year period.
A) Write a linear equation giving the value of the building in terms of the years x after the purchase.
We know that y=75,000 when x=0 (0,75000).
One year after purchase (x=1) the value of the building is $72,500.
So by plugging into the equation, y=mx+b, it will look like
72,500=m*1+75,000.
Therefore, m is equal to and our final linear equation is
y= -2500x
B) In how many years will the value of the building be $42,500?
By plugging into our equation, y= -2500x+75000,
it should look like: 42500=-2500x x=13 years

23. Example: Finding a linear model for Americans’ personal income
From July 1998 to July 1999, Americans’ income rose from 7.13 trillion dollars to 7.50 trillion dollars.
A) Let x=0 represent July 1998, x=1 represent August 1998, etc. Write a linear equation for Americans’ income, y, in terms of the month, x, using the points (0,7.13) and (12,7.5)
By using y=mx+b, we can plug in our points to get 7.5=m* Therefore, m= and our linear equation is y= x+7.13.
B) Estimate Americans’ income in December 1998 (x=5)
y= * … y is approximately 7.28 trillion dollars
C) Predict Americans income in July 2002 (x=48)
y= * … y=8.61 trillion dollars

24. You’re turn!
From July 1998 to July 1999, Americans’ spending rose from 5.82 trillion dollars to 6.20 trillion dollars.
A) Let x=0 represent July 1998, x=1 represent August 1998 and x=12 represents July Write a linear equation (y=mx+b) for Americans’ spending in terms of the month x using the pairs (0,5.82) and (12,6.20).
B) Estimate Americans’ spending in August 1999.
-Keep in mind…what is the x value equal to in this case?
(x=13!)
C) Estimate Americans’ spending in July 2002.
(x=48!)

25. Let’s see how well you did (: Answers
A) By plugging into the equation, y=mx+b, it should look like: 6.2=m* Therefore, m= and the final linear equation is y= x+5.82.
B) By using our formula, y= x +5.82, plug the x value (x=13) into it and solve. y= * … y= trillion dollars
C) By using our formula, y= x +5.82, plug the x value (x=48) into it and solve. Y= * … y= 7.34 trillion dollars