Solving Systems of
Equations

ON THIS PAGE YOU WILL BE TAUGHT HOW TO SOLVE EQUATIONS WITH EASE!!

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There are 3 ways to solve linear equations:
graphing, combination, and substitution!!

Graphing
To graph a two linear equations, first you need 2 equations with at least 2 variables. You then graph the 2 equations by x-y intercepts or T-Chart.

The graph that has only one solution is two lines that cross each other at any point.

The graph with no solution is parallel lines

The graph with infinietly many solutions is two lines that are exacty on top of each other.

3x-2y=6 Equation 1 -> y=3/2x+-3
3x-2y=2 Equation 2 -> y=3/2x+-1

Graph 2 Equations........

Since the 2 equations have the same slope(3/2) the lines are parallel and therefore this system of equations has no solutions.

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You can check it by a calculater(TI-83) or algebraically.

2x-4y=13 Equation 1
4x-5y=8 Equation 2

Lets pick X to eliminate and solve for Y.
Times the equation 1 by -2 to cancel the X's.

-2(2x-4y=13) Distrubute out!!!!
4x-5y=8

-4x+8y=-26 Add them together and then solve!
+4x-5y=8

3y=-18 -> -> {y=-6}

Then put the Y back into one of the original equations and solve for X.

2x-4(-6)=13 -> -> 2x+24=13 -> -> 2x=-11
{x=-11/2}

The solution is {-11/2,-6}

Substitution Method

There are three steps to the substitution method. The first step is solve for one of the
equation's variable. Substitute the expression from step 1 into the the other equation and solve for the
other variable.

Example:

3x+4y=-4 Equation 1
x+2y=2 Equation 2

1.) Solve Equation 2 for X.

X+2y=2 Write equation 2.
X=-2y+2 Revised equation 2.

2.) Substitute the expression for X into equation 1 and solve for Y.

3x+4y=-4 Write equation 1.
3(-2y+2)+4y+-4 Substitute -2y+2 for X.
Y=5 solve for y.

3.) Substitute the value of y into revised equation 2 and solve for x.

x=-2y+2 Write revised Equation 2.
x+-2(5)=2 Substitute 5 for y.
x=8

The solution is (-8,5)

Word Problem

To connect a VCR to a TV set, you need a cable with special connectors at both ends. Suppose you buy a 6 foot cable for $15.50 and a 3 foot cable for $10.50. Assuming that the cost of the cable is the sum of the cost of the 2 connectors and the cost of the cable itself, what would expect to pay for a 4 foot cable?

6x+2y=15.5
3x+2y=10.25 Substract the 2nd equation from the 1st

and you will end up with x=1.75

Then a 4-foot cable will equal 10.25+X

substitute 1.75 for x

The 4 foot cable will cost $12.