# GRAPHING LINEAR EQUATIONS IN TWO VARIABLES

The graphs of linear equations in two variables are straight lines. Linear equations may be written in several forms:

__Slope- Intercept Form :__ y = mx+ b

In an equation of the form y = mx + b, such as y = −2x −
3, the slope is **m** and the y-intercept is the point** (0, b)**. To
graph equations of this form, construct a table of values **( Method 1)** or
use the slope and y-intercept** ( Method 3)** (see Examples 1 and 6).

__General Form:__ ax + by = c

To graph equations of this form, such as 3x − 2y = −6,
find the **x-** and **y-intercepts (Method 2)**, or solve the equation for
y to write it in the form y = mx + b and construct a table of values (see
Example 2).

__Horizontal Lines:__ y = b

The graph of y = b is a horizontal line passing through
the point **(0, b)** on the y-axis. To graph an equation of this form, such
as y = 4, plot the point (0, b) on the y-axis and draw a horizontal line through
it (see Example 4). If the equation is not in the form y = b, solve the equation
for y.

__Vertical Lines:__ x = a

The graph of x = a is a vertical line passing through the
point **(a, 0)** on the **x-axis.** To graph a vertical line, such as 4x +
12 = 0, solve the equation for x to write it in the form **x = a**, plot the
point **(a, 0) **on the x-axis, and draw a vertical line through it (see
Example 5).

__METHOD 1:__ CONSTRUCT A TABLE OF VALUES

To graph equations of the form **y = mx **and **y = mx
+ b,**

1) **Choose** three values for **x**. Substitute
these values in the equation and solve to find the corresponding y-coordinates.

2) Plot the ordered pairs found in step 1.

3) Draw a straight line through the plotted points. If the points do not line
up, a mistake has been made.

__ Example 1:__ Graph y = −2x − 3

To graph the equation, choose three values for x and list
them in a table. (__Hint:__ choose values that are easy to calculate, like
−1, 0, and 1.) Substitute each value in the equation and simplify to find the
corresponding y-coordinate. Plot the ordered pairs and draw a straight line
through the points.

__ Example 2:__ Graph 3x − 2y = −6

The equation 3x − 2y = −6 is written in the general form.
To graph this equation with a table of values, first solve the equation for y to
write it in the form **y = mx + b**, as shown:

Next, choose three values for x and calculate the
corresponding y- coordinates . (__Hint__: to cancel fractions, choose multiples
of the denominator.) Plot the points in the table and draw a line through them.

__METHOD 2:__ FIND THE X- AND Y-INTERCEPTS

In Example 2, the line crosses the x-axis at (−2, 0) and
y-axis at (0, 3). The **point **where the line crosses the** x-axis** is
called the **x-intercept.** At this point, the y-coordinate is **0** The
**point** were the line crosses the **y-axis is** called the **
y-intercept.** At this point, the x-coordinate is **0.**

When an equation is written in the general form, such as −2x + 4y = 8, it is easier to graph the equation by finding the intercepts.

1) To find the **x-intercept**, let **y = 0** then
substitute 0 for y in the equation and solve for x.

2) To find the **y-intercept**, let **x = 0** then substitute 0 for x in
the equation and solve for y.

3) Plot the intercepts, label each point, and draw a straight line through these
points.

** Example 3:** Graph −2x + 4y = 8

1) To graph the equation, find the x- and y-intercepts.

To find the** x-intercept**, let **y = 0** and solve
the equation for x.

The x- intercept is (−4, 0).

To find the **y-intercept**, let **x = 0** and solve
the equation for y.

The y-intercept is (0, 2).

2) Next plot each intercept, label the points, and draw a line through them

**Graphing Horizontal and Vertical Lines**

The graph of **y = b** is a **horizontal line**
passing through the point** (0, b)**, the **y-intercept**. The graph of**
x = a** is a **vertical line** passing through the point **(a, 0)**, the**
x-intercept**.

__ Example 4:__ Graph y = 4

To graph the equation, plot the intercept on the y-axis, label the point, and draw a horizontal line through the point.

__ Example 5:__ Graph 4x + 12 = 0

First, solve the equation for x to write it in the form **
x = a**.

The **x-intercept** is (−3, 0). Plot this point on the
x-axis, label the point, and draw a vertical line through the point.

__METHOD 3:__ USE THE SLOPE AND Y-INTERCEPT

To graph an equation using the slope and y -intercept,

1) Write the equation in the form **y = mx + b** to
find the slope **m **and the y-intercept **(0, b).**

2) Next, plot the y-intercept.

3) From the y-intercept, move up or down and left or right, depending on whether
the slope is positive or negative . Draw a point, and from there, move up or down
and left or right again to find a third point.

4) Draw a straight line through all three points.

** Example 6: **Graph 2x + 5y = 10.

To graph the equation using the slope and y-intercept,
write the equation in the form **y = mx + b** to find the slope **m** and
the y-intercept **(0, b)**.

Now, plot the y-intercept. From there, __move up or down
two units __ (the rise) then __move right or left five units to the right__
(the run) to find additional points.

When the slope is negative, make the change in y negative to locate points to the right of the y-intercept; make the change in x negative to locate points to the left of the y-intercept.

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