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MAT028B
TOPIC: GRAPHS OF LINEAR EQUATIONS
3.1
Graphs and Applications of Linear Equations
(“Linear equations” are quite plain compared to other equations; in two variables,
their lines are straight.)
I. Since you cannot list the infinite number of solutions to an equation with two variables, you
must graph it to show the solutions. (Remember: Solutions make the equation true!)
Ex: x+ y =10 Solutions: (0,10) (10,0) (1/2, 9) (-1, 11) (0.3, 9.7) …
II. The Basic Set Up
Vertical Axis (usually y)
II
This point’s address is
an ordered pair (x,y)
or coordinates.
I
(-3,1)
Origin
III
The marks show scale and should be
consistent on each axis. (They are
worth 1 unless labeled otherwise.)
Horizontal Axis (usually x)
IV
Roman numerals for
quadrants spell a big “C”!
III. The basic way to graph a linear equation like this
Second put x y
x y
Ex: y = 2x + 1
First
Sometimes you
must solve for y first!
make
a#
Table
in some
easy x
values
(at least3)
-1
0
1
x y
Third
Solve -1 2(-1)+1=-1
for 0 2(0)+1=1
each. 1 2(1)+1=3
(“Applications” work the same way, but (x, y) may be replaced by other letters in
alphabetical order.)
The “y-intercept”
happens when x=0
Fourth plot on axes + connect +apply
arrows label
Y=2x+1
x
1
3.2
More with Graphing and Intercepts
I. If your linear equation is not solved for y and looks more like this:
Ex: 2x + 3y = 6
A. Instead of solving for y, try this shortcut
First
make
a#
Table
x
y
Second put
in 0 for each
coordinate
x
Third
solve
for each
in turn
y
0
0
x
y
0 2
3 0
y
Fourth
plot,
connect,
and add
arrows.
2x+3y=6
x
For graphing please use
graph paper and a ruler!
B. The only danger to just using two points. If you would like to “check” pick an easy value
such as 1, substitute it into your equation for one variable, solve for the other, and check to see
that the resulting point is on your line.
C. This shortcut doesn’t work if you only have one variable or a constant of zero.
II. If your linear equation has one variable
Ex: x = 3
Let y vary.
y
Always Vertical
x
3
3
3
y
-1
0
1
x
y
x
Let x vary.
Ex: y = 2
Always Horizontal
Y=2
y
-1 2
0 2
1 2
x
X=3
3.3 Slope and Applications
I. Along with where a line cuts the axes, its “tiltedness” is an interesting feature.
II. The tiltedness of a line is its “slope.” We use “m” as in to mount stairs.
A. Informally: When going from left to right, flat is zero slope, up is positive, down is negative
and freefall is undefined.
B.
More formally: m =
Going from one point to another….
Rising up is positive and sinking is negative.
Running right is positive and left is negative.
C. Quite formally: m =
Ex: Find the slope of the line containing (2,1) (-3,4)
First
assign
(2,1) (-3,4)
Second
plug in #’s
in
either
order!
Third careful
with signs!
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III. Another Shortcut:
If your equation has been solved for you and put in this order
y = mx + b
You can just lift “m” out! It lives next to x when y is alone.
3.4Equations of Lines
I. Finding an equation of a line when the slope and y-int’t are given
In y = mx + b, m =slope and b = y-intercept. So simply substitute them in! (“slope-intercept form”)
II. Finding an equation of a line when slope and a point are given.
Step 1: write y = mx + b
Step 2: Substitute in the slope for m and temporarily put your choice of (x,y) into the x and y in the
y = mx +b
Step 3: Solve for b
Step 4: Write y = mx + b with the given m and the b you found.
III. Finding an equation of a line when two points are given.
Step 1: Find slope using m =
Step 2-5: See II above!
Ex: Given (1,2) (3,-4)
m=
y = mx + b
2 = (-3)(1) + b
b=5
y = -3 x + 5
3.5
Graphing Using the Slope and the Y-Intercept
Trick!
Step 1: Make sure your equation is in slope intercept form (y = mx+b)
Step 2: “Drop anchor” at the y-intercept, leaving a nice dot. (If you don’t see anything, you can add +Ø)
Step 3: Take a look at your slope.
If it’s positive, it’s
Numerator
Denominator (You may put 1 if no denominator is visible.)
If it’s negative, the negative must be attached to the top or the bottom, not both.
Ex:
Step 4: Using the slope, drop in a few more points.
Step 5: Connect the points and add arrows to your line if appropriate.
(P.S. This trick isn’t so great for vertical lines!)
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3.6 Parallel and Perpendicular
I. If slopes are the same and the y-intercepts are same, it’s the same line!
II. If the slopes are the same and y-intercepts are different, the lines are “parallel.”
Ex: y = 2x + 1; y = 2x – 5
III. If the slopes are negative reciprocals (Double Whammy!) the lines are “perpendicular.” Note
that the product of the slopes is -1.
Ex:
IV. Think about vertical and horizontal lines
A. All verticals are parallel with all verticals.
All horizontals are parallel with all horizontals
B. All verticals are perpendicular with all horizontal.
Ex: x = 7 is perpendicular with y = 2.
3.7
Graphing Inequalities in Two Variables
Graphing inequalities is just like graphing equalities, except…
I. Use Dashed line for “strict” inequalities, < >.
Use Solid Line for < , >
II. Shade one side by testing a representative point that is clearly on one side in the original
inequality. The origin is a favorite!
If it comes up “true” shade that side.
If it comes up “false” shade the other.
Ex:
Test
(0,0)
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