Linear Equations
in Two Variables
Linear Equations in Two Variables
may be put in the form
Ax + By = C,
Where A, B, and C are real
numbers and A and B are not
both zero.
Solutions to Linear Equations in Two
Variables

5 x  2 y  20
Consider the equation
 The equation’s solution set is
infinite because there are an
Ordered Pairs
infinite number of x’s and y’s
are listed with the
that make it TRUE.
x-value first and the y-value
second.
 For example, the ordered pair
(0, 10) is a solution because
 Can you list other ordered pairs
that satisfy this equation?
5 0  2 10  20
5 x  2 y  20
 
 
Input-Output Machines


We can think of equations as input-output machines. The
x-values being the “inputs” and the y-values being the
“outputs.”
Choosing any value for input and plugging it into the
equation, we solve for the output.
x=4
y = -2x + 5
y = -2(4) + 5
y = -8 + 5
y = -3
y = -3
Functions

Function- a relationship between two variables
(equation) so that for every INPUT there is
EXACTLY one OUTPUT.
 To determine (algebraically) if an equation is
a function we can examine its x/y table. If it is
possible to get two different outputs for a
certain input- it is NOT a function. In this
case an x-value in the table or ordered pairs
would repeat.
 This may be determined (graphically) by
using the Vertical Line Test. If any vertical
line would touch the graph at more than one
point- it is NOT a function.
Using Tables to List Solutions


For an equation
we can list some
solutions in a table.
Or, we may list the
solutions in ordered
pairs .
{(0,-4), (6,0), (3,-2),
( 3/2, -3), (-3,-6),
(-6,-8), … }
2 x  3 y  12
x
y
0
-4
6
0
3
-2
3/2
-3
-3
-6
-6
-8
…
…
Graphing a Solution Set
To obtain a more complete picture of a
solution set we can graph the ordered
pairs from our table onto a rectangular
coordinate system.
 Let’s familiarize ourselves with the
Cartesian coordinate system.

Cartesian Plane
y-axis
Quadrant II
( - ,+)
Quadrant I
(+,+)
x- axis
Quadrant III
(-,-)
Quadrant IV
(+, - )
Graphing Ordered Pairs on a Cartesian Plane
y-axis
1) Begin at the origin
2) Use the x-coordinate
to move right (+) or
left (-) on the x-axis
3) From that position
move either up(+) or
down(-) according to
the
y-coordinate
(6,0)
x- axis
4) Place a dot to indicate
a point on the plane
Examples: (0,-4)
(6, 0)
(-3,-6)
(0,-4)
(-3, -6)
Graphing More Ordered Pairs from our
Table for the equation
y
2 x  3 y  12
•Plotting more points
we see a pattern.
•Connecting the points
a line is formed.
x
(3,-2)
•We indicate that the
pattern continues by placing
arrows on the line.
•Every point on this line is a
solution of its equation.
(3/2,-3)
(-6, -8)
Graphing Linear Equations
in Two Variables





The graph of any linear
equation in two variables
is a straight line.
Finding intercepts can be
helpful when graphing.
The x-intercept is the
point where the line
crosses the x-axis.
The y-intercept is the
point where the line
crosses the y-axis.
On our previous graph,
y = 2x – 3y = 12, find the
intercepts.
y
x
Graphing Linear Equations
in Two Variables
y
On our previous graph,
y = 2x – 3y = 12, find
the intercepts.
The x-intercept is (6,0).
x
The y-intercept is (0,-4).
Finding INTERCEPTS

To find the
x-intercept: Plug in
ZERO for y and solve
for x.
2x – 3y = 12
2x – 3(0) = 12
2x = 12
x=6
Thus, the x-intercept is (6,0).
To find the
y-intercept: Plug in
ZERO for x and solve
for y.
2(0) – 3y = 12
2(0) – 3y = 12
-3y = 12
y = -4
Thus, the y-intercept is
(0,-4).
Special Lines
y+5=0
x=3
y = -5
y
y
x
x
y = # is a horizontal line
x = # is a vertical line

SLOPE- is the rate of
change
 We sometimes think
of it as the steepness,
slant, or grade.
Slope formula:
y y2  y1 rise
slope  m 


x x2  x1 run
Slope:
Given 2 colinear points, find the slope.
Find the slope of the line
containing (3,2) and (-1,5).
y2  y1
25
3
m


x2  x1 3   1 4
Slopes
Positive slopes rise from left to right
Negative slopes fall from left to right
Special Slopes

Vertical lines have
UNDEFINED slope
(run=0 --- undefined)
 Horizontal lines have
zero slope (rise = 0)
 Parallel lines have the
same slope (same
slant)
 Perpendicular lines
have opposite
reciprocal slopes
m  undefined
m0
m1  m2
1
m1  
m2
Slope-Intercept Form
y = mx + b
where m is the slope and b
is the y-intercept
Graph using Slope-Intercept form
Given:
2y= 6x – 4
y = 3x – 2
nd
Plot (0, -2) then use 3/1 as rise/run to get 2
y point:
n
n
n
4.
Solve for y.
Plot b on the y-axis.
Use m  rise
run to plot
a second point.
Connect the points to
make a line.
1
x
3
b
Rise: positive means UP/ negative means DOWN
Run: positive means RIGHT/ negative means LEFT
Determine the relationship between
lines using their slopes
 y  3x  2

2

3
y

9
x

2

y

3
x


3

Are the lines parallel, perpendicular or
neither?
 Solve for y to get in Slope-Intercept form.
 Then compare slopes.

Determine the relationship between
lines using their slopes
 y  3x  2

1

3
y

x

6

y

x

2

3

Are the lines parallel, perpendicular or
neither?
 Solve for y to get in Slope-Intercept form.
 Then compare slopes.
Write an Equation
given the slope and y-intercept
Given: That a line passes through (0,-9)
and has a slope of ½ , write its
equation.
 (0,-9) is the y-intercept (because x=0)
 ½ is the slope or m
 Plug into the Slope Intercept Formula to

get:
y= ½ x - 9
Point-Slope Form

At times we may not know the y-intercept.
Thus, we need a new formula. The pointslope form of a line going through x1 , y1
with a slope of m is given by

y  y1  m  x  x1 

Use Point-Slope
when you don’t have a y-intercept

Given two points (1,5)
and (-4,-2), write the
equation for their line.
 Choose one point to
plug in for (x1,y1)
 Find the slope using
both points and the
slope formula.
 Solve the equation for y.
y  y1  m  x  x1 
y  5  m( x  1)
7
y  5   x  1
5
7
18
y  x
5
5
Modeling Data with Linear
Equations
•Data can sometimes be
modeled by a linear
function.
•Notice there is a basic
trend. If we place a line
over the tops of the
bars it “roughly fits.”
Each bar is close to the
line. Thus points on the
line should estimate our
data.
•Given the equation to
the line we can make
predictions about this
data.
Female Enrollment
NW-SCC
1990 1994 1998 2000
NW- 2378 2785 3052 3245
SCC
Modeling Data with Linear
Equations
•The number of U.S.
children (in
thousands) educated
at home for selected
years is given in the
table. Letting x=3
represent the year
1993, use the first
and last data points
to write an equation
in slope-intercept
form to fit the data.
Home Schooling in U.S.
x
y
3
588
4
735
5
800
6
920
7
1100
y=128x + 204