Graphing Method

A system of two linear equations in two variables x and y consists of two equations of the following form:

Equation 1: Ax+By=C

Equation 2: Dx+Ey=F

A solution of a system of linear equations in two varibles is an ordered pair (x,y) that satisfies each equation.

Note: There are 3 ways to answer a linear equation, Infinitely Many Solutions, No Solution, or Exactly One Solution.

Infinitely Many Solutions are solutions that occur when the equations are the same line.

No Solution means that the equations will never intersect.

Exactly One Solution means the equations will intersect at exactly one point.

Example:

Tell how many solutions the linear system has.

3x-2y=6

6x-4y=12

Step 1: Change the equation to where the y is by itself.

Step 2: If needed divide every number in the equation with the number in front of the y, for there to be only one y.

Step 3:Graph the equation by slope-intercept form

The answer to the Example above is Infinitely Many Solutions.

Note: You can check your answer by plugging in your answer into a TI-83/TI-84 calculator and graphing it.

Substitution Method

Substitution is one way to solve linear system algrbraically.

Example:

3x+4y=-4

x+2y=2

Step 1: Solve one of the equations for one of its variables.

Step 2: Substitute the expression from Step 1 into the other equation and solve for the other variable.

Step 3: Substitute the value from Step 2 into the revised equation from Step 1 and solve.

The answer to the Example above is (-8,5)

Note: You can check the solution by substituting the answers back into the original equations.

Linear Combination Method

Linear Combination is another way to solve linear systems algrbraically.

Example:

2x-4y=13

4x-5y=8

Step 1: Multiply one or both of the equations by a constant to obtain coefficients that differ only in sign for one of the variables.

Step 2: Add the revised equations from Step 1. Combining like terms will eliminate one of the variables. Solve for the remaining variable.

Step 3: Substitute the value obtained in Step 2 into either of the original equations and solve for the other variable.

The answer to the Example above is (-11/2,-6)

Note: You can check the solution by using a graphing calculator or substituting the answers back into the orginal equations.

Word Problem

To connect a VCR to a television 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 $10.25. Assuming that the cost of a cable is the sum of the cost of the two connectors and the cost of the cable itself, what would you expect to pay for a 4 foot cable? Explain how you got your answer.

3x+2y=10.25

x=cost per foot of cable

y=cost of one connector

Subtract the second equation from the first and solve for x.

x=1.75

Plug the answer into one of the equations and solve for y.

y=2.50

4(1.75)+2(2.50)=12

The cost of a 4 foot cable is $12.