Download 6.2 Graphing Linear Equations Equations with only variable and

6.2 Graphing Linear Equations
Equations with only variable and with no exponents have
only one solution.
E.g. 2x – 1 = -5
The only solution is x=-2
However, linear equations, which are equations with two
variables and no exponents, have an infinite number of
solutions.
How about 2x - 3y = 6 ?
What values of x and y make this equation true?
Well, you could have x=- and y = -2 [2(0) – 3(-2) = 6]
Or you could have x=-3 and y = -4 [2(-3) – 3(-4) = -6+12 = 6]
And so on and so on. There are too many to name. We can
state the solution by graphing it.
How do you graph an equation like 2x – 3y = 6?
4
y
3
2
1
0
-4
-3
-2
-1
-1 0
-2
-3
-4
-5
x
1
2
3
4
Linear equations have a dependent variable and an independent variable. In
general, x is the independent variable and y is the dependent variable. When
a linear equation is re-written so that y is on one side and x is on the other,
the value of y depends on the value chosen for x.
How to graph a linear equation:
Step 1) Re-arrange the equation so that y is isolated.
2x - 3y = 6
+2x
+2x
-3y = 2x + 6
(even though we added 2x to 6 it is “proper” to
place the variable term before the constant term.)
Now divide both sides by -3 to get y by itself.
− 3y − 2x 6
=
+
−3
−3 −3
2
y = x−2
3
Step 2) Set up a table of ordered pairs and choose x’s, then find out what
the y’s need to be. You only need two ordered pairs to draw a line, but three
helps it to be more accurate. When choosing x’s, try to pick ones that will
come out with an integer
x
y=⅔x-2
(x,y)
when multiplied by the coefficient
-3 =⅔(-3)-2=-2-2=-4 (-3,-4)
in your linear equation.
Now that you have three ordered
pairs, just plot them on a graph and
connect the dots!
0
=⅔(0) -2=-2
(0,-2)
3
=⅔(3)-2=2-2=0
(3,0)
Sometimes it’s hard to choose x’s that will give you a nice integer
for y. An alternative method is using the slope and intercept of the
line.
Graph 2y – x = 2
Get y by itself by first adding x to both sides
2y - x + x = 2 + x
2y = x + 2
Divide both sides by 2
2y x 2
= +
2 2 2
1
y = x +1
2
Once y is isolated, it is now in what is called the “slope-intercept form.”
y = mx + b, where m represents something called the slope and b
represents the y-intercept.
The slope of the line tells you how much it is slanted. The y-intercept is
the point where the line crosses the y-axis. At that point x=0 and y is the
Run 2 units
constant.
y
2
In this case, the slope is ½
and the y-intercept is 1
[or more specifically (0,1)]
A slope of ½ means that from any
point on the line, the line
rises vertically 1 unit and
“runs” horizontally 2 units.
Rise 1 unit
1
x
0
-2
-1
0
1
-1
-2
We know right away that one point on the line is the y-intercept: (0,1).
So we can start at that point and rise 1 unit and then run 2 units to land
at another point on the line (2,2).
2
Slopes tell you immediately about the steepness and direction of a line.
A slope that is positive points up and to the right
A slope that is negative points down and to the right.
If the rise is negative, the vertical rise in a downward direction.
If the run is negative, the horizontal “run” is to the left.
−2 2
=
1
−1
down 2 units up 2 unit
=
= same " slant"
right 1 unit
left 1 units
A slope of - 2 =
A slope whose absolute value is a fraction less than 1 is closer to being
horizontal.
A slope whose absolute value is greater than one is steeper and closer to being
vertical
Graph
y = 3x – 2
This is already in slope-intercept form.
The slope is ____
The y-intercept is (0,__)
y
Run 1 unit
6
5
4
Rise 3 units
3
2
1
0
-6
-5
-4
-3
-2
-1
x
0
-1
-2
-3
-4
-5
-6
1
2
3
4
5
6