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 Depdendent Variable

 Number of equations to solve: 23456789
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 Dependent Variable

 Number of inequalities to solve: 23456789
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 Solve for:

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# Graphing Linear Equations

A linear equation is an equation in two variables, each raised to the 1st power

Ex: is linear, is linear, is not linear, nor is .

The solution of a linear equation is an ordered pair that makes the equation true.

Ex. 1
Is the point (5, 2) a solution of ? (In other words, does the line formed by pass through the point (5, 2)?)

To find out, substitute 5 (the x-coordinate) for the x and the 2 (the y-coordinate) for y. Does the
left side equal the right side? Since the ordered pair made the equation true, the ordered pair is a solution of the linear
equation.

Is the ordered pair (9, 0) a solution of x + 2y = 9? Yes, this ordered pair is also a solution. So is
the ordered pair (-7, 8). A linear equation will have an infinite number of solutions.

There are three methods of graphing linear equations that we will use: 1) making a table, 2) using
x and y intercepts and 3) using the slope-intercept method.

Ex. 2
Graph by making a table.

1. y = x + 2

First, pick three values for x (you have to choose at least two x values but you can choose more
than three). They can be any values you want. I usually pick zero, one positive number and one
negative number: To find the y-values, substitute each of the x values into the equation y = x + 2. So the y value
that corresponds to x = 0 is y = 0 + 2, or 2. The y value that corresponds to x = 1 is 3 and the y
value that corresponds to x = -2 is 0. So I'll complete my table: Each of these x and y values make an ordered pair. So the line y = x + 2 passes through the
points (0, 2), (1, 3), and (-2, 0). Now graph each one of those ordered pairs and connect them
with a line to get this graph: I would like to note here that you might want to pick a little more choosy on your x values when
the coefficient of your variable x is a fraction. You can choose any value you want but you can
choose x values that will give you integer y values. A good rule of thumb to follow is this: when the coefficient of the x term is a fraction, choose
zero, the denominator and its opposite. Obviously, substituting a zero for the x gives us or y = 0 + 3. When we put in the

3, notice what happens: , two-thirds of 3 is 2. So y = 2 + 3 or y = 5. If we had
used a 1 for x (and you can use any number that you want), you would have gotten which would give you . While it is fine to graph a
fraction, it also means that you have to approximate. Graphing integers will give you a more
precise line. So back to the table, when all of the x values have been substituted, your table
should look like this: And the graph will look like this: There are a couple of special case to point out here. Suppose you wanted to graph the lines y = 4
and x = 4. You might argue that these are not linear equations since there appears to be only one
variable. But keep in mind that y = 4 can also be written as 0x + y = 4 (also y = 0x + 4) and x = 4
can be written as x + 0y = 4. So the graphs of these two equations are a bit different than the
others. If we make a table and choose 3 values for x and substitute those values in, our table
would look like this: Now graphing those points would give us a graph
that looked liked this: Note: The equation y = k (where k is any constant) is a horizontal line that crosses the y axis at
k. And so it stands to reason that the equation x = k is a vertical line that crosses the x axis at k.
So x = 4 will look like this: Now we'll graph using the x and y intercept method. By definition, the x intercept is the point
where the line crosses the x-axis. This will always be where y = 0 so the x intercept will be an
ordered pair (x, 0). The y intercept is the point where the line crosses the y-axis. This will
always be where x = 0 so the y intercept will be an ordered pair (0, y). We'll use those definitions
to graph the next example.

Ex. 3
Graph using x and y intercepts.

1. 3x + 2y = 6
*The first thing that you may notice is that the form of this linear equation
is different from the previous example. The x and y are on the same side 