
It is possible to solve systems of equations using a table.
The tables may be drawn in a variety of forms and shapes, including tables generated from a graphing calculator.
The table method is a "partner" to the graphical solution.
A table of values is a listing of select points from the straight line graphs.
Where the lines intersect on a graph, the yvalues will be the same.
In the tables, we will be looking for the xlocation where the yvalues are the same.
Example 1:
Solve the system y = x  1 and y = 2x + 5
using a table.
A horizontal table is shown at the right,
containing the two linear equations and a listing of possible x values.
The solution is (x, y) = (2, 1) 


In a table solution, xvalues are substituted into both linear equations in the system. When the same yvalue appears in BOTH equations, the solution has been found. 


When working with a table solution method,
place both linear equations in "y =" form.
This will make it eeasier to examine the yvalues.


Example 2:
Solve the system x + y = 6 and 2x + y = 8
using two separate tables.
First, put the equations into "y =" form:
y = x + 6
y = 2x + 8
The vertical tables are shown at the right,
containing the two linear equations and a listing of possible x values.
The solution is (x, y) = (2, 4) 

Example 3:
Solve the system x  y = 3 and 2x + y = 6
using the graphing calculator table.
First, put the equations into "y =" form:
y = x + 3
y = 2x + 6
The graphing calculator table is shown at the right, containing the two linear equations and a listing of possible x values.
The solution is (x, y) = (1, 4) 
Let Y_{1} = x + 3 and Y_{2} = 2x = 6.
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The screens below do not need to be displayed in a table solution, but they show the equations used to create the table, and that the intersection point (the solution) is at the point (1, 4). 




When using the graphing calculator, you will still need to set the equations equal to y before you begin.
Follow the link at the right to see how to use your graphing calculator with systems and how to see the table, and control the table view.




For calculator help with systems of equations
click here. 


What happens with much larger (or much smaller) integer solutions?
If the xvalue in the solution is an integer from 4 and +4, it is fairly easy to test each of those 9 values to see which might be the solution. But what if the answer is not an integer in that range? What if the answer is an xvalue that is much larger (or smaller)? Is there a way to narrow in on what that value might be?
ANSWER: Let's assume that the xsolution is still an integer, but it is not in the range from 4 to 4. While there is no way to "immediately" find the xsolution using just a table, you can narrow in on the solution, by looking at the difference between the yvalues. The difference between the yvalues should be getting smaller if you are heading toward the solution. Remember that when using a graph to find the solution to a system, the point of intersection of the lines is the answer. At the point of intersection, the difference between the yvalues is 0.
Example 4:
Solve the system y = x + 8 and y = 2x  4 using a table.
Let's start our table in the positive x direction from 0 to 4.
No solution yet! Notice that the difference between the yvalues is getting smaller, so we are heading in the right direction, but we need to jump ahead faster (pick some larger values for x). 
Since the difference of the yvalues was decreasing by 1 each time, we can count that we would need 8 more chart entries to get the difference to be 0. Eight more chart entries puts the xvalue at 12. Solution (12, 20) 
Example 5:
Solve the system y = x + 8 and y = 2x + 20 using a table.
Let's again start our table in the positive x direction from 0 to 4.
Stop!! The difference in the yvalues is getting larger. We are heading in the wrong direction to arrive at the answer. We need our xvalue to be going in the negative direction.
Change the direction to have xvalues from 0 to 4, for starters.
The difference in the yvalues shows we are headed in the correct direction, but we need to move further to the left. Solution: (12, 4)
At the beginning of this page, it was stated that tables are "partners" with graphs.
The examples above show some of the problems you can encounter when solving a system of equations using only the table method. These examples all had nice integer answers.
The problems encountered with the table method and rational solutions (including decimals and fractions) arise very quickly. If, however, you "partner" the table method with a graph (even a hand drawn graph), you will have an indication of the approximate location of the intersection (the answer), and you can adjust the table accordingly to narrow in on the solution.
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