Solutions to Linear Equations in Two Variables
1. Given 2x − 3y = 6 build the table for using an arithmetic sequence as replacement values.
→ Logically, we know that it is easy to plot ordered pairs if
both coordinates are integers.
In this equation we note that all of the numbers except the ycoefficient are
even integers.
From arithmetic we know that:
Multiplying by any even integer will yield
an even integer as the product
and
Adding two even integers will result in
an even integer as the sum.

Even Ã— Odd (or Even) = Even
Even Â± Even = Even 
Now if we choose an even integer as the replacement for the y value then the resulting xvalue
will be an integer. Therefore, we put the "sequence of even integers" in the table for y
then complete the computation below for each yvalue and put each corresponding
xvalue in
the table. Repeat the process for the other yvalues to find the resulting
xvalues. Note any
sequences and write the common differences.
2x − 3y = 6
Let y =  4: 2x − 3( 4) = 6,
repeat for the other numbers
y =  2: 2x − 3( 2) = 6,
y = 0: 2x − 3( 0 ) = 6,
y = 2: 2x − 3( 2 ) = 6,
y = 4: 2x − 3( 4 ) = 6, 
x =  3
x = 0
x = 3
x = 6
x = 9 
Check to see that
both columns of values are arithmetic sequences.

For the point ( 3,  4) replace x =  3 and y =  4 in the given equation 2x − 3y = 6
CHECK:
For the point (9, 4) replace x = 9 and y = 4 in the given equation 2x − 3y = 6
CHECK:
You should always form the practice of checking all of your work whenever
you can.
2. Graph 2x + 3y = 7. Build a table for 2x + 3y = 7.
Again let us turn to LOGIC: We know that for integers:
Odd Ã— Odd = Odd
Odd Â± Odd = Even 
Even Ã— Odd (or Even) = Even Even Â± Odd = Odd 
In order to keep both coordinates as integers choose the replacement
values for the y to be odd integers. We choose the sequence of odd integers for
replacement for y values and the resulting xsequence will be integers. Complete the
computation below and put the number in the table. Repeat the process for the other yvalues to
find the resulting xvalues.
2x + 3y = 7 Let y = 1
repeat for the other numbers

Put these values in
the table, then write
the â€œcommon
differencesâ€ for both
x and y.
Use the value for dx
to complete the table
values for x.

For complete table see below. 
Note adding Â± 3 to given values forms an arithmetic sequence.
To be sure you should always check the â€œouter pointsâ€.


Check:
Check:
Check the dy and dx on your table.
