5.5 Standard Form:
X-intercept: The point where the graph
crosses the x-axis, ( y=0).
Y-intercept: The point where the graph
crosses the y-axis, (x=0).
Linear Equation: is an equation that
models a linear function.
GOAL:
Whenever we are given a graph we must be able
to provide the equation of the function in
Standard Form: The linear equation of the
form:
Ax + By = C
where A, B and C are real whole numbers (no
fractions) and A and B are not both zero.
EX:
What are the x- and y-intercepts of the
graph of 5x – 6y = 60?
SOLUTION: There are many ways to find this
information depending on the form you are
given, but if you are given the standard form
(Ax+By=C), then you must plug in zero for
the other variable.
Finding the x-intercept: plug in zero for y
5x – 6y = 60
plug in y=0
5x – 6(0) = 60
5x = 60
x = 60/5  12 (12,0) is the point.
Finding the y-intercept: plug in zero for x
5x – 6y = 60
plug in x=0
5(0) – 6y = 60
– 6y = 60
y=60/-6  -10 (0, -10) is the point
Graph:
𝟓𝒙 − 𝟔𝒚 = 𝟔𝟎
X-intercept:
(12, 0)
2
Y-intercept:
( 0, -10)
-2
-2
2
YOU TRY IT:
What are the x- and y-intercepts of the
graph of 3x + 4y = 24?
YOU TRY IT: (SOLUTION)
Finding the x-intercept: plug in zero for y
3x + 4y = 24
3x + 4(0) = 24
plug in y=0
3x = 24
x = 24/3  8
(8,0) is the point.
Finding the y-intercept: plug in zero for x
3x + 4y = 24
3(0) + 4y = 24
plug in X=0
4y = 24
y = 24/4  6
(0,6) is the point.
Graph:
X-intercept:
(8, 0)
Y-intercept:
( 0, 6)
3𝒙 + 𝟒𝒚 = 𝟐𝟒
Graphing Horizontal Lines
Remember: x lines are vertical
y lines are Horizontal
X=3
y=-2
YOU TRY IT:
What are the graphs of
x = -1 and y = 5
YOU TRY IT: (SOLUTION)
Remember: x lines are vertical
y lines are Horizontal
X = -1
y=5
TRANSFORMING TO STANDARD FORM
If we are given an equation in slope-intercept
from (y = mx +b), and the point-slope form
(y – y1=m(x-x1)) we can rewrite the equations
into standard form:
Ax + By = C
where A, B and C are real whole numbers (no
fractions) and A and B are not both zero.
EX:
What are the standard forms of
𝟑
𝟏
1) y = - x + 5 and 2) y – 2 = - (x + 6)
𝟕
𝟑
SOLUTION: 1)
𝟑
Using the slope-intercept from y = - x + 5
𝟕
We must get rid of any fraction, no fractions
allowed:
𝟑
𝟕
y=- x+5
Inverse of dividing by 7
7y = - 3x + 35
Inverse subtraction 3x
7y + 3x= 35
Variables in order
3x + 7y = 35
Ax + By = C form.
Graph:
X-intercept:
(11.7, 0)
Y-intercept:
(0, 5)
Here we
would use:
𝟑
y=- x+5
𝟕
down 3,
right 7
𝟑𝒙 + 𝟕𝒚 = 𝟑𝟓
SOLUTION: 2)
𝟏
Using the point-slope from y-2 = - (x + 6)
𝟑
We must first distribute the slope
𝟏
𝟑
y -2 = - x - 2
Distribute -
𝟏
𝟑
We must then get rid of fractions
Inverse of division by 3
3y - 6 = - x -6
(multiply everything by 3).
3y + X = -6 +6
Variables to left numbers to
the right of equal sign.
x + 3y = 0
Ax + By = C form.
Graph:
𝒙 + 𝟑𝒚 = 𝟎
X-intercept:
(0, 0)
Y-intercept:
(0, 0)
We now use
𝟏
y=- x+0
𝟑
USING STANDARD FORM AS MODEL
In real-world situations we can write and
use linear equations to obtain important
information to help us find out what we
can do with the resources we have.
EX:
In a video game, you earn 5 points for
each jewel you find. You earn 2 points for
each star you find. Write and graph an
equation that represents the number of
jewels and stars you must find to earn
250 points.
What are three possible combinations of
jewels and stars you can find that will
earn you 250 points?
SOLUTION:
In a video game, you earn 5 points for each
jewel you find.
Let x = the jewels you find.
You earn 2 points for each star you find.
Let y = the starts you find.
Write the equation for a total of 250 points:
5x + 2y = 250
Graph:
5𝒙 + 𝟐𝒚 = 𝟐𝟓𝟎
X-intercept:
(50, 0)
250
Stars
225
200
Y-intercept:
(0, 125)
175
150
125
100
75
50
25
25 50 75 100 125
Jewels
Graph:
Three points are:
(0, 125)
0 Jewels,
125 Stars
(25, 63)
25 Jewels,
62.5 Stars
225
200
Stars
(25, 62.5)
25 Jewels,
62.5 Stars
250
175
150
125
100
75
50
25
25 50 75 100 125
Jewels
VIDEOS:
Graphs
https://www.khanacademy.org/math/algebra/line
ar-equations-and-inequalitie/point-slopeform/v/linear-equations-in-standard-form
https://www.khanacademy.org/math/algebra/line
ar-equations-and-inequalitie/point-slopeform/v/point-slope-and-standard-form
CLASSWORK:
Page 323-325
Problems: As many as
needed to master
the concept