System of LINEAR Equations : Two Equations

Recall: A system of equations is a collection of equations with common variables.

Recall: The solutions to a system of equations are the sets of values for the variables which make
every equation true .

If a system of equations has two variables , the solutions can be stated as ordered pairs.
If a system of equations has three variables, the solutions can be stated as ordered triples .

Recall: Linear equations of two variables are equations whose graph can be represented by a line.
The equation can be represented where every variable has an exponent of either 1 or 0.

Nonlinear equations of two variables are equations that aren’ t linear .

Recall: There are four methods to solving systems of linear equations: finding intersection points of
the graph, the method of substitution , the method of elimination, and applications of matrices.

Solving Systems of Equations by graphing:

If the appropriate graphs of two equations can be sketched, then the solutions can be found by
identifying the ordered pair coordinates of each point of intersection. If the equations can be
solved for y , it is possible to ENTER the equations into a calculator and use the graphing tools to
find the point of intersection.

Steps for a TI -82, 83, 83+, 83+SE, or 84:

Press the Y = button and ENTER your equations in the equation editor. Select a window
that will appropriately graph each of the equations . Press the GRAPH button and look at
the graph. If intersection points do not appear and the graphs do not imply intersection
points, you can probably stop. If intersection points do not appear but the graph implies
that intersection points may occur outside your window, then adjust your window. If
intersection points appear, press 2nd and TRACE to bring up the “CALC” menu and
press 5 for “intersection”. Use the left and right arrow buttons to move your cursor along
the graph of the first equation to the point of intersection and then press ENTER. You will
then be shifted to the second equation. Press ENTER twice more and the point of
intersection should appear as coordinates along the bottom of your screen.

Steps for a TI -85 or 86:

Press the GRAPH and F1 buttons to open your equation editor—ENTER your equations.
Select a window that will appropriately graph each of the equations. Press the GRAPH
and F5 buttons to look at the graph. If intersection points do not appear and the graphs
do not imply intersection points, you can probably stop. If intersection points do not
appear but the graph implies that intersection points may occur outside your window,
then adjust your window. If intersection points appear, press the MORE, F1 (MATH),
MORE, and F3 (ISECT) buttons. Use the left and right arrow buttons to move your
cursor along the graph of the first equation to the point of intersection and then press
ENTER. You will then be shifted to the second equation. Press ENTER twice more and
the point of intersection should appear as coordinates along the bottom of your screen.