Please
CLOSE
YOUR LAPTOPS,
and turn off and put away your
cell phones,
and get out your notetaking materials.
A linear equation in two variables:
• Is of the form Ax + By = C
• A, B, C are real numbers
• A & B are not both zero. (It’s OK for one or the
other to be zero, just not both at the same time.)
This is called the “standard form” of a linear
equation.
• “Standard form” has the x and y terms on the
left and the constant (number) term on right,
and usually, all fractions cleared away by
multiplying by LCD.
• (We’ll also be studying two other forms of a
linear equation: “slope-intercept” form and
“point-slope” form.)
Graphing linear equations:
• Find at least 2 points on the line.
• Connect the points to form a line.
Example: Graph the linear equation 2x - y = -4.
Process: Find two ordered pair solutions (and a third solution as
a check of the computations) by choosing a value for one of the
variables, x or y, then solving for the other variable. We plot the
solution points, then draw the line containing the 3 points.
Example (cont.)
Graph the linear equation 2x - y = -4.
(NOTE: You can pick any number you want for x, so pick something simple and
easy, like 0 or 1 or 2 or -1 or -2.)
Let x = 1.
Then 2x - y = -4 becomes
2(1) – y = -4
2 – y = -4
(replace x with 1)
(simplify the left side)
-y = -4 – 2 = -6
y=6
(subtract 2 from both sides)
(multiply both sides by -1)
So one solution is (1, 6).
Example (cont.)
Graph the linear equation 2x - y = -4.
For the second solution, let y = 4. (Could be any number you choose.)
Then 2x - y = -4 becomes
2x – 4 = -4
2x = -4 + 4
2x = 0
x=0
(replace y with 4)
(add 4 to both sides)
(simplify the right side)
(divide both sides by 2)
A second solution is (0, 4).
Example (cont.)
Graph the linear equation 2x - y = -4.
For the third solution, let x = -3. (Could be any number you choose.)
Then 2x - y = -4 becomes
2(-3) – y = -4
-6 – y = -4
(replace x with -3)
(simplify the left side)
-y = -4 + 6 = 2
y = -2
(add 6 to both sides)
(multiply both sides by -1)
A third solution is (-3, -2). (Note: different people would
probably pick different numbers and get three different points, and
they’d still come out with the same graphed line.)
Now we plot all three of the solutions (1, 6), (0, 4) and (-3, -2).
y
And then we
draw the line
that contains
the three
points.
(1, 6)
(0, 4)
x
(-3, -2)
Helpful Hint
When graphing a linear equation in two
variables:
• if it is solved for y, it may be easier to find
ordered pair solutions by choosing xvalues.
• If it is solved for x, it may be easier to find
ordered pair solutions by choosing yvalues.
Example
3
Graph the linear equation y = 4 x + 3.
Since the equation is solved for y, we should
choose values for x.
To avoid fractions, we should select values of x
that are multiples of 4 (the denominator of the
fraction).
(The online graphing tool usually does not allow you to
graph fraction coordinates, so you need to select values of x
and y that will make both coordinates integers.)
Example (cont.)
3
Graph the linear equation y = 4 x + 3.
Let x = 4.
Then y =
3
4
x + 3 becomes
y=
3
4
(4) + 3
(replace x with 4)
y=3+3=6
One solution is (4, 6).
(simplify the right side)
Example (cont.)
3
Graph the linear equation y = 4 x + 3.
For the second solution, let x = 0. (Zero is always a good,
easy-to-graph number to pick for one of your values.)
Then y =
3
4
x + 3 becomes
y=
3
4
(0) + 3
y=0+3=3
(replace x with 0)
(simplify the right side)
A second solution is (0, 3).
Example (cont.)
3
Graph the linear equation y = 4 x + 3.
For the third solution, let x = -4.
Then y =
3
4
x + 3 becomes
y=
3
4
(-4) + 3
y = -3 + 3 = 0
(replace x with -4)
(simplify the right side)
A third solution is (-4, 0).
Now we plot all three of the solutions (4, 6), (0, 3) and (-4, 0).
y
And then we
draw the line
that contains
the three
points.
(4, 6)
(0, 3)
(-4, 0)
x
Question:
Can we classify an equation like x = 5 or y = -11
as a “linear equation in two variables”?
Recall the definition of a linear equation in
two variables:
• It is of the form Ax + By = C
• A, B, C are real numbers
• A & B are not both zero. (It’s OK for one or the
other to be zero, just not both at the same time.)
So back to the question:
• Can we call x = 5 and y = -11 “linear equation in
two variables”?
• What would their graphs look like in 2 variables?
Example: Graph the linear equation y = 3 on a twovariable graph grid.
• Notice that the equation can be written in standard
form as 0x + y = 3.
• No matter what value we replace x with, y is
always 3.
x
y
0
3
1
3
5
3
Besides linear equations, we will also
examine the graphs of quadratic
equations, absolute value equations and
other nonlinear equations in later sections.
Non-linear simply means the graph is not
a straight line.
•
•
One type of a non-linear equation is the quadratic.
A quadratic has ONE of the variables squared (we’ll
use x as the squared variable for examples in this section).
•
The graph of a quadratic equation is shaped like a “U”
with the ends either pointing upwards (like a cup) or
downwards (like a hill.)
•
To graph a quadratic equation, plug values in the
equation for the x and calculate the corresponding y values
to get the ordered pairs to plot as the points on your graph.
•
You might need a LOT of points – go until you find
where the graph reaches a turning point and starts curving
back in the other direction.
Example
x
y
2
4
1
-2
0
-4
-1
-2
-2
4
Graph y = 2x2 – 4.
y
(-2, 4)
(2, 4)
x
(-1, -2)
(1, -2)
(0, -4)
Application problem from today’s homework
that can be solved by graphing:
The assignment on this material (HW 3.2)
Is due at the start of the next class session.
Lab hours:
Mondays through Thursdays
8:00 a.m. to 6:30 p.m.
Please remember to sign in
on the Math 110 clipboard
by the front door of the lab
You may now OPEN
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and begin working on the
homework assignment.
We expect all students to stay in the classroom
to work on your homework till the end of the 55minute class period. If you have already finished
the homework assignment for today’s section,
you should work ahead on the next one or work
on the next practice test.