# Graphing Linear Equations in Two-Variables

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```					3.3                          Graphing Linear Equations in Two-Variables                           page 14
Build a table of ordered pairs where the x-values are in a sequence and the y-values are in a
sequence. When you plot these points (ordered pairs) you will see that they form a straight line.

Always choose only an integer as the replacement for the x-value so that the resulting y-value
will be an integer, and visa versa. If we put a “sequence of integers" in the table for x , complete
the computation for each x-value, and put each corresponding y-value in the table, then the
y-values in the table will be a “sequence of integers". Write the common differences.

1. Given 2x − 3y = 6 build the table for using an arithmetic sequence as replacement values.

Note:   In this equation if you choose consecutive integers for x you will find some y-values that
are fractions. By noting that the constant 6 is a multiple of 3 and the coefficient of y is 3, you
can choose your sequence for the x-values to be multiples of 3 . Now, when you are solving
for the y-values and divide by the coefficient-3 you will obtain a sequence of integers for the
y-values.

x     y
2x − 3y = 6
-3    -4
dx = 3      0    -2       dy= 2
Let y = - 4:   2x − 3(- 4) = 6,       x= -3
3     0
6     2
repeat for the other numbers                                                9     4

Check to see that both columns
of values are arithmetic sequences.

y

Start at the leftmost point and count the                                                           •
blocks up to the line of the next point.
•
Ratio: dy       2                         •           x
=                           •
dx       3
Now count the blocks right to the next                                              •
point.
Compare these to dy and dx.

See the next page for another approach to graphing this equation.
page 15                           Graphing Linear Equations                                               3.3

2. Build the table for 2x − 3y = 6 using an arithmetic sequence as replacement values.

For another approach:

If both the x-intercept and the x-intercept are integers, we can place them in the middle of the
table. Then we can find the difference in the x-values for dx and the difference in the y-values for
dy and use these differences to build arithmetic sequences for the table.

Let x = 0, then   2( 0 ) − 3y = 6 or y = -2                                 x      y
-3     -4
which gives the point: ( 0 , -2 )                                  -3                      -2
0     -2
dx= 3       3      0       dy= 2
Let y = 0, then   2x − 3( 0 ) = 6 or x = 3
6      2
+3                      +2
which gives the point: ( 3 , 0 )                                            9      4

Check the dy and dx on your table.                                                dy          2
Ratio:      =        3
dx

Now you must CHECK the top and bottom points in the table to be sure that they are points on
the line:

For the point (- 3, - 4) replace x = - 3 and y = - 4 in the given equation 2x − 3y = 6
?
CHECK:               2(- 3 ) − 3(- 4 ) = 6

- 6 + 12       = 6     !
For the point (9, 4) replace x = 9 and y = 4 in the given equation 2x − 3y = 6
?
2( 9 ) − 3( 4 ) = 6
CHECK:                     18 − 12   = 6       !

Students should always form the practice of checking all of their work whenever they can.
page 16                                Graphing Linear Equations                                              3.3
3. Graph 2x + 3y = 7.

Look at the coefficients and note that if we choose a sequence of odd integers as replacement
for y-values , such as: { -3, -1, 1, 3, 5}, the resulting x-sequence will be integers

Build a table for        2x + 3y = 7.
Put these values in the
middle of the table, then
2x + 3y = 7        Let y = 1            write the “common                                x   y
differences” for both x                             -3
2x + 3( 1) = 7     or    x=2            and y .                                             -1
dx= - 3          2   1 dy= 2
Use the value for dx to                          -1 3
complete the table                                   5
repeat for y = 3                        values for x.

2x + 3( 3) = 7,     or    x =-1         Note the value for dy in
the table values for y.       For complete table see below.

Note adding ± 3 to given x-values                                                   x   y
forms an arithmetic sequence.                                                       8 -3
+3
5 -1  -2
Note adding m 2 to given y-values
dx= - 3          2   1 dy= 2
forms an arithmetic sequence.
-1 3
-3                   +2
-4 5
To be sure you should always check the “outer points”.

?
Check: (- 4, 5) " 2( - 4) + 3( 5 ) = 7 or       - 8 + 15 = 7     !

?
Check: (8, - 3) " 2( 8 ) + 3( - 3 ) = 7 or + 16 – 9 = 7          !
Plot the points, draw line:

y

Check the dy and dx on your graph.                                                      •
Ratio:
dy
=
-2                -2        •
dx     3                      +3       •            x
- 2•
•
+3
page 17                              Graphing Linear Equations                                                  3.3

For other examples study the coefficients for combinations that will yield integer points (x1, y1).

3. 3x + 5y = 7

Since 10 − 3 = 7, choose x = -1
and y = 2 to satisfy this difference.

3( -1) + 5( 2) = 7           (-1, 2)

Repeat for the another pair of numbers

Since 5 + 7 = 12, choose y = -1
and solve for x.                            Put these values in the                      x        y
middle of the table,
3x + 5( -1) = 7                             then write their
3x − 5 = 7                  (4, -1)         “common differences”
+5      +5                                                         dx= + 5          -1        2 dy=- 3
for as dx and dy .
3x = 12 or          x = 4                                                           4        -1
Use the value for dx to
complete the table
dy - 3          3                            values for x.
=       =                                                                              dy        -3
dx   5         -5                                and                             Ratio:      =
dx            5
dy to complete the
table values for y.
Note adding m 5 to given x-values
x  y
NOTE: Opposite signs                         -11 8
and ± 3 to given y-values forms                 above given values.          -5                      +3
-6  5
an arithmetic sequence for each.                                            dx= + 5          - 1 2 dy=- 3
4 -1
+5                      -3
9 -4
To be sure you should always check the “outer points”.

Check: (- 11, 8) " 3( - 11) + 5( 8 ) ? 7 or
=             - 33 + 40 = 7   !      Plot the points, draw line:

?                                                             y
Check: (9, - 4) " 3( 9 ) + 5( - 4 ) = 7        or + 27 – 20 = 7    !
•
-3
+5
•
•
-3                                                     x
•
dy
Check the dy and dx on your graph.             Ratio:    =
dx       5                                      -3
+5
•
page 18                                    Graphing Linear Equations                                                                   3.3
Three special types of lines:

1. Horizontal lines, y = b            2. Vertical lines, x = a              3. Lines through the origin, y = mx

1. Horizontal lines, y = b      (any real number)                                              (- 6, 2), (- 3, 2), (0, 2), (3, 2), (6, 2)
Equation: y = 2
Horizontal (y = b)          (y is the same for every              x-value)

Every point on a horizontal line has the same second number, for all x-values.

Example 1: Graph the line for the                                      Example 2: Given two points with the same
equation: y = - 3                                                second number: (- 7, 5), (2, 5)
Equation: y = 5
y                                                                      y

(- 7, 5)         (2,
•               •
5)

x                                                                      x
(0, - 3)
•

2. Vertical lines, x = a     (any real number)
(3, - 4), (3, - 2), (3, 0), (3, 2), (3, 4)
Equation: x = 3
Vertical (x = a) For every y-value.

Every point on a vertical line has the same first number, for all y-values.

Example 1: Graph the line for the                                 Example 2: Given two points with the same
Equation x = 4.                                                   second number: (- 7, 5), (- 7, - 3)
Equation: x = - 7                               -8
y                                                             y
(- 7, 5), (- 7, - 3)

0
(- 7, 5)     •

•             x                                                               x
(4, 0)
(- 7, - 3)    •
page 19                                 Graphing Linear Equations                                                3.3

3. Lines through the origin: y = m x               ⇒ m is the coefficient, any real number , times x.

If a line goes through the origin then (0, 0) is one point on the line, ⇒ y = m x

The Line is:                                          The coefficient is:                     Example:

Rising as x moves from left to right                       m > 0, (Positive)

Falling as x moves from left to right                      m < 0, (Negative)

⇒ Given an equation in the form y = mx where there is no constant, it is equal to zero, the line
will always go through the origin (0, 0).

Example 1: Equation: y = 5/2 x,         m>0                 Example 2: Equation: y = - 2/3 x,           m<0

Find and plot the points (- 2, - 5), (0, 0), (2, 5),        Find and plot the two points (- 3, 2), (0, 0), (3, - 2)

Draw the line through the points.                           Draw the line through the points.

y                                                      y

•(2,                                        (- 3, 2)
5)
•
x                                                        x
• ( 3, - 2)
•
(- 2, - 5)

```
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