The Substitution Method

Step 1:
Solve one of the equations for one of it's variables.

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

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

In other words......

Equation
1: 6x + y = -2
2: 4x - 3y = 17

Revise the Equation
y = -6x - 2

Step 1: Substitute Equation 1 for Y.
4x - 3(-6x - 2) = 17
4x + 18x + 6 = 17
X = 1/2

Step 2:
Substitute X back into Equation 1.
y = -6x - 2
y = -6(1/2) - 2
y = -5

The Solution is ( 1/2, 5)

The Linear Combination Method

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

Step 2:
Add the revised equations from Step 1 and solve.

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

In other words.....

Equation
6x + 3y = 3
8x + 4y = 4

Step 1
24x + 12y = 12 ( Multiply equation 1 by 4)
-24x - 12y = -12 (Multiply Equation 2 by -3)
0 = 0 ( Add the equations)

Step 1:
Put both equations in " y = mx + b "

Step 2:
Graph both equations as it is on the right.

Step3:
Find the intersection. That is your solution.

In other words...


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


Graph

WORD PROBLEM

Your family is planning a 7 day trip to Flordia for the Flordia - Georgia game. You estimate that it will cost $275 per day for your first stop and $400 per day at your second stop. Your total budget for the 7 days is $2300. How many days should you spend in each location?

Labels:

Equation 1
Time spent at the first location = x
Time spent at the second location = y
Total vacation time = 7

Equation 2
Daily rate in Tampa = 275
Time spent in Tampa = x
Daily rate in Orlando = 400
Time spent in Orlando = y
Total budget = 2300

Equation 1: x + y = 7

Equation 2: 275x + 400y = 2300

Find out what
x + y = 7

(4) + (3) = 7

So......

275 (4) + 400 (3) = 2300