Transcript 10.3 Slope-Intercept Form of Linear Equations

Slope-Intercept Form of
Linear Equations
Objectives
• Write an equation in slopeintercept form given the slope and
y-intercept,
• Determine the slope and yintercepts of a graph, and
• Determine the x- and y-intercepts
of a graph
Application
• Larry is taking home economics. As
a class project, he is cooking dinner
for some friends. Larry plans to
spend $15 on beef and/or chicken.
If beef costs $5 per pound and
chicken costs $3 per pound, the
graph at the right shows all the
possible combinations of beef and
chicken he can buy. If he buys no
beef, he can buy 5 pounds of
chicken. If he buys no chicken, he
can buy 3 pounds of beef. The
number 5 on the x-axis and the
number 3 on the y-axis are called
intercepts.
More on intercepts
• The x coordinate of the point
where a line crosses the x-axis
is called the x-intercept.
• The y coordinate of the point
where the line crosses the yaxis is called the y-intercept.
Consider the graph below
• The line with slope m
crosses the y-axis at (0,
b). You can write an
equation for this line using
the point-slope form. Let
(x1, y1 ) = (0, b)
y – y1 = m(x – x1)
y – b = m(x – 0)
y = mx + b
Slope-Intercept Form
• Given the y-intercept b and slope
m of a line, the slope-intercept
form of an equation of the line is
y = mx + b
Ex. 1: State the slope and y-intercept of the
graph of y = 5x – 3
y = 5x – 3
y = mx + b
Since m = 5, the slope is
5. Since b is -3, the yintercept is -3.
Compare the slope-intercept form of a linear equation with
the standard form, Ax + By = C. Solve for y.
Ax  By  C
By   Ax  C
B
 Ax  C
y
B
B
A
C
y  x
B
B
If the equation is given in
standard form, and B is not
zero, the slope of the line is 
The y-intercept is
C
B
A
B
Ex. 2: State the slope and y-intercept of the
graph of 3x + 2y = 12
• This is easy
given that all the
numbers are
positive. BE
CAREFUL!
Negatives can
really mess you
up here.
• In the equation 3x +
2y = 12, A = 3; B = 2
and C = 12
A
3
 
B
2
C 12

6
B 2
Ex. 2: State the slope and y-intercept of the graph
of 3x + 2y = 12—CHECK YOUR ANSWERS!
• Write the equation in
slope-intercept form.
3x  2 y  12
2 y  3x  12
3 12
y 
2 2
3
y   6
2
A
3
 
B
2
C 12

6
B 2
The slope is -3/2 and the yintercept is 6. The solution
checks.
Recall
• Recall that the x-coordinate of the
ordered pair for the y-intercept is 0
and the y-coordinate of the ordered
pair for the x-intercept is 0. You can
use these facts to find the x- and yintercepts of the graph of a linear
equation.
Ex. 4: Determine the x- and y-intercepts of
the graph of 4x – 5y = 10
To find the x-intercept, let y = 0.
4x – 5y = 10
4x – 5(0) = 10
4x = 10
x = 10/4
x = 5/2
To find the y-intercept, let x = 0.
4x – 5y = 10
4(0) – 5y = 10
-5y = 10
y = 10/-5
y = -2
The x-intercept is 5/2. The graph
crosses the x-axis at (5/2, 0).
The y-intercept is -2. The graph
crosses the x-axis at (0, -2).
Formula for the x-intercept
• The y – intercept of the graph
of an equation in the form Ax +
By = C is C .
B
• To find the formula for the xintercept, let y = 0
• The x intercept then is C/A, A ≠
0. Therefore, for the graph of
2x – 3y = 8, the x-intercept is
8/2 or 4, and the y-intercept is
– 8/3
Ax  By  C
Ax  B (0)  C
Ax  C
C
x
A