Procedures/Intro for Solving Systems of Linear Equations

The system we will use has two equations with two variables in each equation.

ax	+	by	=	h
cx	+	dy	=	k

This substitution method uses two equations, in which one is solved for x or y and then pluged-in to the second equation and solved for x or y depending on what variable was solved in the first equation.

The posible answers for this method are:
1. one point in common (two intersecting lines)
2. infinite points in common (two paralel lines occupying the same space)
3. no points in common (two paralel lines)

The following is an example of a system that is solved by solving for y and substituting.
The first equation:

2x - y = 1

The second equation:

3x + 2y = 12

1. Solve for y in the first equation. The first equation is equal to:

2x - 1 = y

2. The second equation's y-value is replaced by 2x-1.

3x + 2(2x-1) = 12

3. Simplify the equation:

7x - 2 = 12

4. Simplify the equation:

7x = 14

5. Simplify the equation:

x = 2

6. Substitute 2 for x in the first equation:

2(2) - 1 = y

7. Simplify the equation:

3 = y

8. Evaluate the answer.
a) The unique solution is made by x=2 and y=3, thus the intersection point of the two systems is (2,3).
b) If the answer was 0=0, then the two linear equations would occupy the same space and would have infinitely many points in common.
c) If the answer was wrong (i.e. 0=4), then the two linear equations would have no common point and would be parallel to each other.

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