MA 12A
Arcadia Valley Career Technology Center
Topic: Linear Equations and Systems
Show-Me Standards: MA1,MA4
Tech Math I Credit
Last Update: March 2008
Focus: Identifying Linear
Equations/Graphing Linear Equations
MO Grade Level Expectations: : N2C8,
N2C9, A2B7
NCTM Standards: 4A, 5A
OBJECTIVE: The students will be able to identify a linear equation and graph an equation
using T-tables and intercepts.
Terms:
Linear Equation – has no operations other than addition, subtraction, and multiplication of a
variable by a constant number. The variables may not be multiplied together or appear in a
denominator. It only contains variables taken to a power of 1. The graph of the generated pairs is
always a line.
Linear Equations
Non-Linear Equations
x+y=4
7x2 + y = 4
5x – 2y = 9
y = √ x – 11
y = -3
x + xy =5
x = 10
y = 2/x
4a = 2b – 7
4ab = 12
y=⅜x
|m|=n
Y –Intercept – the y-coordinate of the point where the line crosses the y-axis. Denoted by the letter
“b” .
X - Intercept – the x-coordinate of the point where the line crosses the x-axis.
T-table – a chart to generate points on the line.
Examples:
State whether the following are linear equations. If not, explain why.
3x – y = 12
yes
xy = 4
no, variables are
multiplied together
y2 = x + 7
no, variable taken
to power other
than 1
x/y = 3
no, variable in
denominator
y=½x+3
yes
y=9
yes
x=6
yes
x = 5y
yes
GRAPHING LINEAR EQUATI0NS: T-tables, Intercepts, and Horizontal/Vertical
Using T-tables:
To graph linear equations, we can generate pairs of coordinates that satisfy the equation.
We use a T-table to help us generate the pairs.
Example:
Graph 6x – 2y = 10 using a T-table.
-To start, it is usually helpful to solve the equation for the y variable. (In other words, get
the y variable all alone on one side of the equal sign).
6x – 2y = 10
6x – 6x – 2y = 10 – 6x
-2y = 10 – 6x
-2y = 10 – 6x
-2
-2 -2
y = -5 + 3x
, note 6x – 2y = 10 and y = -5 + 3x represent the same line.
-Now, pick some random values for x . 1,-1,0 and any other numbers that may be
appropriate are our usual picks. Organize your picks in a T-table.
X
1
-1
0
4
Y_
y = -5 + 3(1) = -2
y = -5 + 3(0) = -5
X
1
-1
0
4
Y_
-2
-8
-5
7
y = -5 + 3(-1) = -8
y = -5 + 3(4) = 7
so the pairs generated are (1,-2),(-1,-8),(0,-5),(4,7)
-Now graph the pairs and connect them to form the line.
y
^
● (4,7)
● (1,-2)
●
●
>
x
(0,-5)
(-1,-8)
-there are thousands of pairs that will connect to this line. We chose four to get a
look at the direction and steepness of the line.
Using Intercepts:
To graph using the x and y intercepts, we let x = 0 and y = 0 in our T-table. This
method can be quicker since we only have to form two pairs.
Example:
Graph 3y = 12 – 4x using intercepts.
X
0
Y_
0
3y = 12 – 4(0)
3y = 12
y=4
3(0) = 12 – 4x
0 = 12 – 4x
x=3
X
0
3
Y_
4
0
so the pairs that locate the intercepts of the line are
(0,4) and (3,0).
-Now graph the intercept pairs.
y ^
● (0,4)
●
(3,0)
>
x
Horizontal and Vertical Lines:
Horizontal and Vertical lines are the easiest to graph because they are the easiest to
identify. No T-table is necessary. Horizontal lines always come in the equation form of
y = any number and vertical lines always come in the form x = any number .
Example:
Graph y = 3 and x = -2 .
-
y = 3 tells us that no matter what x is, y will always be 3. All pairs will have
3 as their y-coordinate.
x = -2 tells us that no matter what y is, x will always be -2. All pairs will have
-2 as their x-coordinate.
- Now graph.
y
^
y ^
●
●
●
●
●
●
●
y=3
●
●
>
x
●
>
x
●
x = -2
Problems:
Tell whether the following equations are linear equations. If not, explain why.
1. 7x + 5y = 19
2. 7/y = 4x
3. 5 – y = 3
4. 2xy + 6 = y
5. 4x2 – 3x – 5 = 8
6. x = 23
7. 4a3 + b = 7
8. x = ⅓ y
9. y = ½ x – 3
10. | k | + j = 8
Graph the following equations using a T-table.
11. y – 3x = -2
12. 4y + 8 = -16x
Graph the following using the intercepts of the line.
13. 5x + 7y = 35
14. 2y = 3x – 6
Graph the following horizontal and vertical lines.
15. y = -5
16. x = 6
17. 4½ = y
18. x = 0