Linear Functions
A Linear function is a function in the form of:
F(x)=mx + b
Or
Function also known as
Y= mx + b
The two terms ‘m’ and ‘b’ are traditional (they are fixed numbers)
The role of M
If y = mx + b , then:
(a) Whatever the term M is, multiplying it by the X term then changes it.
(b) Solving for m, we obtain m by:
m =y
x
= change in y
change in x
Role of b: When x = 0, y = b or f(0) = b
Examples
The function
F(x) = 5x-1
In this linear function the m = 5 and b = -1.
The following equations can be solved for y as linear function of x like so:
1) 3x –y +4 = 0
y = 3x + 4
2) 4y = 0
4y = 0
4
y=0
3 ) 3x + 4y = 5
4y = 5 + -3x
4
y = 5/4 + -(3/4)x
There are many steps in linear function one in which is collecting your data and make a
data table
(0
(1
(2
(3
(4
, 1)
, 3)
, 5)
, 7)
, 9)
these are you order pairs
Step two create a data table by taking your order pairs and place
them into the x and y points ( terms ) as shown.
x y
0 1
1 3
2 5
3 7
4 9
The next step in the development a linear function is the graph, which has a horizontal
axis that will represents the x points. A vertical axis represents the y points in your Data
table.
Ex: of your pairs in graph form
The fourth step in linear function is to calculate the slope of the line on the graph. This is
done in the following way:
1. You pick any two points on the in your graph with the ( x,y )
2. Calculate the distance the line goes up from the two points that you have chosen.
do this by counting the spaces up.
3. Do the same for the distance sideways . After that you divide they
x
known as the change in y over the change in x. This will find your slope.
And the second y – the first y; the same for the x’s, the first x – the second x
Ex: find the slope of the line that pass through ( 3 ,2 ) and ( 2 , 5 )
5–2
2–3
= equals
3
-1
= equals
-3
There is the graph that we just found the slope of the line of.
The step that I have mentioned works for lines with positive slope. If your line is a
negative slope, then take the negative of the ratio between the two distances. If you use
this formula the sign will automatically be correct.
Ex: 2
8–0
0 -5
=
8
-5
When you are given two equations, you can find where the two lines will intercept and
find for the common ordered pairs. There are two methods for solving this. You can use
the graphing method, or you can use the algebic method.
The graphing method is simple, you just graph the two equations in y = mx + b, and
record where they intercept.
With a data table
X
0
1
2
3
4
5
6
7
8
9
Y=4x + 2
2
6
10
14
18
22
26
30
34
38
X
0
1
2
3
4
5
6
7
8
9
Y=3x+10
10
13
16
19
22
25
28
31
34
37
The highlighted spot are the intercepting point.
Solving with algebra.
Y = 4x + 2
Y = 3x + 10
Substitute y with 3x + 10
3x + 10 = 4x + 2
-3x + 3x + 10 = 4x – 3x + 2 Subtract 3x from each side
Remember what you do to
one side you do to the other
10= x + 2
10 – 2 = x +2 – 2
8=x
We wanted to know what x equaled so we isolated it from the other terms.
You then can find y by simply placing the value of x in the unknown term ‘x’, in one of
these equations.
Y = 3x + 10
Y = 3(8) + 10
Y = 24 +10
Y = 34
Substitution
8x + 3y = 41
150x + 80y = 840
Rearrange one of the two equations to show the form y = mx + b the substitute it for y.
150x + 80y = 40
150x + 80( -8x + 41)= 840
3
150x + -640 = 840 – 3280
3
3
631/3x = -2531/3
x=4
You ,then take that x =4 and place it in a equation to find the valve of y
8(4) +3y=41
32-32 + 3y = 41-32
3y = 41-32
3y = 9
3
3
y=2
Elimination
8x + 3y + 41 -150(8x-3y = 41) –1200x + 450y = 6150
150x + 80y + 840(150x + 80y = 840) 1200x + 640y = 6720
190y = 570
190 190
y=3
Then take the three and put it back in the equation to get the x = value
8x + 3y = 41
8x + 3(3) = 41
8x = 41 – 9
8x = 32
x=4
For the following equations you will need to fill in x or y with zero to find the unknown
term.
Example: 5x – 2y = 10
Solution:
Let x = 0.5(0) –2y = 10
-2y = 10
y = -5
(0,-5)
Let y = 0.5x – 2(0) = 10
5x = 10
x=2
(2,0)
Let x = 4.5940 –2y = 10
20 – 2y = 10
-2y = -10
y=5
(4,5)
Letting x = 0 will give us the y-intercept
Solve for y.
Letting y = 0 will give us the x-intercept
The third point is a check
These points would be plotted and a line drawn through them to complete the graph.
EXERCISE QUESTIONS
Solve for x and y using zeros
1) 5x + 10y = 30
2) 3x + 9y + -18
3) –2x + 8y + 24 = 0
Solve for y = mx + b
4) 5x + 10y + 20 = 0
5) 9x –3y = -6
6) –6x + 4y – 12 = 0
Substitution
7) x = 2y – 1 & 2x + y = 3
8) x = 2y – 8 & x = 1 – y
9) x = 3y + 1 & 3x + y = 13
10) x = y + 2 & 2x – 3y = 2
Elimination
11) x= 2y + 4
x = 3y – 2
12) 3x + 4y = 5
5x – 4y = -13
13) 6x + 2y = 6
-9x + 2y = 9
Graph the following
14) y = 4x + 12
15) 2/3y = x - 6
EXERCISE ANSWERS
1) 5x + 10y = 30
5(0) + 10y = 30
10y = 30
10 10
y=3
5x + 10(0) = 30
5x = 30
5
5
x=6
2) -3x + 9y = -18
–3(0) + 9y = -18
9y = -18
9
9
y = -2
-3x + 9(0) = -18
-3x = -18
-3
-3
x=6
3) -2x + 8y + 24 = 0
-2(0) + 8y + 24 = 0
8y = -24
8
8
y = -3
-2x + 8(0) + 24 = 0
-2x = -24
-2
-2
x = 12
4) 5x + 10y + 20 = 0
5x + 10y = -20
10y = -5x –20
10
10
y = -1/2x - 2
5) 9x – 3y = -6
-3y = -9x -6
-3
-3
y = 3x + 2
6) -6x + 4y –12 = 0
-6x + 4y = 12
4y = 6x + 12
4
4
y = 3/2x + 3
7) x = 2y – 1 & 2x + y = 3
2(2y – 1) + y = 3
4y – 2 + y = 3
5y – 2 = 3
5y = 5
5 5
y=1
x = 2y -1
x = 2(1) - 1
x=1
8) x = 2y – 8 & x = 1 – y
2y –8 = 1 – y
3y –8 =1
3y = 9
3 3
y=3
9) x = 3y + 1 & 3x + y = 13
x = 3(1) + 1
x=3+1
x=4
x = 2y – 8
x = 2(3) – 8
x=6–8
x = -2
3(3y + 1) + y = 13
9x + 3 + y = 13
10y+ 3 = 13
10y = 10
10
10
y=1
10) x = y + 2
x=2+2
x=4
2x – 3y = 2
2(y + 2) + 3y = 2
2y + 4 + 3y = 2
-1y + 4 = 2 – 4
-1y = -2
-1
-1
y=2
11) x = 2y + 4
x = 3y -2
2y + 4 = 3y - 2
2y – 3y + 4 = -2 –4
2y – 3y = -2 –4
-1y = -6
-1
-1
y=6
x = 2(6) + 4
x = 12 + 4
x = 16
12) 3x + 4y =5
5x – 4y = -13
8x
= -8
8
8
x = -1
3(-1) + 4y = 5
-3 + 4y = 5
4y = 8
4 4
y=2
13) 6x + 2y = 6
-9x + 2y = -9
-3x
=-3
-3
-3
x=1
9(1) + 2y = 9
9 + 2y = 9
2y = 0
2
2
y=0
14)
15)
Now that these questions have been completed, you’re ready for the real thing.
Here is a mini quiz, do the following questions.
Find the distance between the following ordered pairs with the slope.
1) (0,5) & (1,10)
2) (-4, 6) & (2, 10)
3) (-2, 15) & (3, 20)
Find x and y intercepts for each of the following.
4) 3x + 2y = 12
5) 5x + 10 = 30
Rearrange each of the following into y = mx + b form.
6) 2y = 6x - 2
7) .6x + .2y – 4 = 0
8) 5y – x/4 = 1
9) 3y + x – 6 = 7 – 2x
10) 4x – 3y + 2x = y - 7
Answers for the mini quiz.
1) y2 – y1 = 10 – 5 = 5 = 5
X2 – X1
1–0 1
2) 20 – 15 = 5 = 5
3–2
1
3) 10 – 6 = 4 = -2
2 - 4 -2
4) 3x + 2y = 12
3(0) + 2y = 12
2y = 12
2
2
y=6
5) 5x +10y = 30
5(o) + 10y= 30
10y = 30
10
10
y=3
6) y + 3x - 1
7) 7 = -3x + 20
8) y = 1/20x + 1/5
9) y = -x + 13/3
10) y = 3/2x + 7/4
For further information see these sites:
http://faculty.tamu-commerce.edu/webesp/557-005/tellez/
http://www.mohawkc.on.ca/dept/math/kezys/6061linear.PDF
http://www.vcsun.org/~bsamii/schedule/notes/mod9/linearfunctions.html#2 .
Bibliography:
1. http://www.ste2000.org/math/linear.htm
2. http://id.mind.net/~zona/mmts/functionInstitute/linearFunctions/linearFunctio
ns.html
3. http://www.cut-the-knot.com/do_you_know/linear.shtml