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Solving Systems of Linear Equations Using Linear Combination
An interior design company placed two orders with an invention/landscaping company. The first order was for 13 bushes that water themselves and 4 trees and totaled $487. The second order was for 6 more bushes of the same and 2 more trees and totaled $232. The bill does not list the per-item price. What is the cost for each bush and each tree?
Solution....
Pick variables and set up a system of linear equations.
Here are the equations that you will be working with.	13b+4t=487 (Equation 1) 6b+2t=232 (Equation 2)
13b+4t=487 6b+2t=232	Solve the linear system.
13b+4t=487 -2(6b+2t)=232(-2)	Multiply the second row so that one of the variables will have the opposite value of the same variable in the other equation. You will need to do this so that you can get rid of one of the variables.
13b+4t=487 -12b-4t=-464	When you multiply the second row, you get....
b=23	Add the two equations together to solve for b.
13(23)+4t=487	Substitute the value of b back into the first equation to find the value of t.
299+4t=487 4t=188 t=47	Solve for t.
As you can tell, b=23 and t=47. This means that each one of the bushes costs $23 and each of the trees costs $47.

Solving Systems of Linear Equations By Graphing
Solve the following system by graphing.		2x-3y=-2 (Equation 1) 4x+y=24 (Equation 2)
Solution....
2x-3y=-2 -3y=-2x-2 y=(2/3)x+(2/3)	4x+y=24 y=-4x+24	Solve each equation for y.
		Graph the first line by taking the number that is not "connected" with the x and using that number as the y-intercept on the graph. Then, take the number that is "connected" with the x and use that number as the slope of the line.
		Then do the same thing that you did for the first line and graph it on the same graph as you used for the other one. The place in which the two lines cross is the solution to the linear system.
The solution to the system is (5,4).

Solving Systems of Linear Equations Using Substitution
Solve the linear system.	x+y=1 (Equation 1) 2x-3y=12 (Equation 2)
Solution....
Solve for y in Equation 1.
y=-x+1	Revised Equation 1.
Substitute -x+1 for y in Equation 2 and solve for x.
2x-3y=12	Write Equation 2.
2x-3(-x+1)=12	Substitute -x+1 for y.
2x+3x-3=12	Distribute the -3.
5x-3=12	Simplify.
5x=15	Add 3 to each side.
x=3	Solve for x.
To find the value of y, substitue 3 for x in the revised Equation 1.
y=-x+1	Write revised Equation 1.
y=-3+1	Substitue 3 for x.
y=-2	Solve for y.
The solution in (3, -2).