Section 7.1

This section is all about solving systems of equations and the different ways it can be done!

There are 3 main ways to solve a system of equations...
1) Substitution
2) Elimination
3) Graphing

Substitution
Example:
x + y = 4
2x + 1 = y

By plugging the second equation into the first we can conclude that:

x + (2x+1) = 4
3x + 1 = 4
3x = 3
x = 1

Some simple steps for substitution include...
1. Solve one equation for one variable in terms of the other
2. Substitute the expression found in step 1 into the other equation to obtain an equation in one variable
3. Solve the equation obtained in step 2
4. Back- substitute the solution in step 3 into the expression obtained in step 1 to find the value of the other variable
5. Check that the solution satisfies each of the original equations

Elimination
Example:
3x + 5y =10
2x - 5y = 5

Given this set of equations, we can elimate the 5y and -5y and solve the equation...

3x = 10
2x = 5
_______
5x = 15
x = 3

Remember a few easy steps when solving a system of equations using elimination:
1. Obtain coefficients for x (or y) that differ only in sign by multiplying all terms of one or both equations by suitably chosen constants
2. Add the equations to eliminate one variable; solve the resulting equation
3. Back- substitute the value obtained in Step 2 into either of the original equations and solve for the other variable
4. Check your solution in both of the original equations

Graphing
Graphing to find the solution of a system of equations comes in handy when the set is too challenging to solve algebraically. It should not be a first resort when solving the system though. When you graph the two equations, remember the solution is where the two graphs intersect!