2. 1. A 72 ft. pipe is cut into two pieces of lengths in a 5:7
ratio. What are the lengths of the two pieces?
2. A $40.00 shirt has been discounted 35%. What is the sale
price of the shirt?
3. Find a number so that 20 more than one-third of the
number equals three-fourths of that number.
4. x + 2 + x - 1 = 2
3
2
5. x + 3 = x - 2
6
4

3. Formulas:
The Slope of a Line:
The Slope Intercept Form:
The slope of the line passing
through the two points (x1,
y1) and (x2, y2) is given by
the formula
A linear equation written in
the form y = mx + b is in
slope-intercept form.

4. Equations and their Graphs:
Equations of the form ax + by = c are called linear
equations in two variables.
y
Graph the equation 2x + 3y = 12.
Find the x - intercept.
Find the y-intercept.
x
-2
2
4

5. 5

6. The slope of the line passing through the two
points (x1, y1) and (x2, y2) is given by the formula
y
(x2, y2)
y2 – y1
change in y
(x1, y1)
x2 – x1
change in x
x
6

7. Example: Find the slope of the line passing
through the points (2, 3) and (4, 5).
Use the slope formula with x1= 2, y1 = 3, x2 = 4, and y2 = 5.
y2 – y1
5–3
m=
=
x2 – x1
4–2
2
=1
=
2
y
(4, 5)
2
(2, 3)
2
x
7

8. The Slope Intercept Form of a Line:
A linear equation written in the form y = mx + b is in
slope-intercept form.
The slope is m and the y-intercept is (0, b).
To graph an equation in slope-intercept form:
1. Write the equation in the form y = mx + b. Identify m and b.
2. Plot the y-intercept (0, b).
3. Starting at the y-intercept, find another point on the line
using the slope.
4. Draw the line through (0, b) and the point located using the
slope.
8

9. Example: Graph the line y = 2x – 4.
1. The equation y = 2x – 4 is in the slope-intercept form.
So, m = 2 and b = - 4.
y
2. Plot the y-intercept, (0, - 4).
x
3. The slope is 2. m =
change in y
2
=
1
change in x
4. Start at the point (0, 4).
(0, - 4)
Count 1 unit to the right and 2 units up
1
to locate a second point on the line.
The point (1, -2) is also on the line.
(1, -2)
2
5. Draw the line through (0, 4) and (1, -2).
9

10. The Slope Intercept Form of a Line:
Example 2: Graph the line 3y - 3x = 6.
Last Example: Graph the line -2x + y = -4.
slope
intercept
y = 2x - 4.
When you know how to 'decode' the slope-intercept form, you
won't have to make a table of x and y values any longer.

13. Class Work:
See Handout

14. A linear equation written in the form y – y1 = m(x – x1)
is in point-slope form.
The graph of this equation is a line with slope m
passing through the point (x1, y1).
Example:
The graph of the equation
y
8
m=-
y – 3 = - 1 (x – 4) is a line
2
of slope m = - 1 passing
2
through the point (4, 3).
4
1
2
(4, 3)
x
4
8
14

15. Example: Write the slope-intercept form for the equation
of the line through the point (-2, 5) with a slope of 3.
Use the point-slope form, y – y1 = m(x – x1), with m = 3 and
(x1, y1) = (-2, 5).
y – y1 = m(x – x1)
Point-slope form
y – y1 = 3(x – x1)
Let m = 3.
y – 5 = 3(x – (-2))
Let (x1, y1) = (-2, 5).
y – 5 = 3(x + 2)
Simplify.
y = 3x + 11
Slope-intercept form
15

16. Example: Write the slope-intercept form for the
equation of the line through the points (4, 3) and (2, 5).
5–3 =- 2 =- 1
m=
-2 – 4
6
3
y – y1 = m(x – x1)
1
(x – 4)
3
y = - 1 x + 13
3
3
y–3=-
Calculate the slope.
Point-slope form
Use m = -
1
and the point (4, 3).
3
Slope-intercept form
16

17. Two lines are parallel if they have the same slope.
If the lines have slopes m1 and m2, then the lines
are parallel whenever m1 = m2.
y
(0, 4)
Example:
The lines y = 2x – 3
y = 2x + 4
and y = 2x + 4 have slopes
m1 = 2 and m2 = 2.
x
y = 2x – 3
The lines are parallel.
(0, -3)
17

18. Two lines are perpendicular if their slopes are
negative reciprocals of each other.
If two lines have slopes m1 and m2, then the lines
are perpendicular whenever
y
1
m2= or m1m2 = -1.
y = 3x – 1
m1
(0, 4)
Example:
The lines y = 3x – 1 and
1
y = - x + 4 have slopes
3
m1 = 3 and m2 = -1 .
3
1
y=- x+4
3
x
(0, -1)
The lines are perpendicular.
18

19. • Linear Equations form straight lines. How do we
determine if an equation is linear?
It can be rewritten in the form: Ax + By = C
This is the Standard Form of a linear equation where:
a.) A and B are not both zero.
b.) The largest exponent is not greater than 1
Determine Whether the Equations are Linear:
1. 4 - 2y = 6x
2. -4/5x = -2
3. -6y + x = 5y - 2
Remember:
This is to determine whether an equation is linear
(forms a straight line) or not. The standard form is also
used to determine x and y intercepts.

20. Class Work:
8-3: 28-36;
Plot & Label 3 points per problem
4.3-- All
20