1. Solving Systems of Linear Equations using Elimination

2. Elimination
Any system of linear equations in two variables can be solved by the elimination method – also called the addition method.
The first trick is to get the equations lined up so that the same variables are in a column. Sometimes the problem comes that way, other times you will need to “rearrange the furniture” before you can start.

3. Elimination
2x + 3y = 42
2x + 4y = 50
The object of the game is to get one column to add to zero.
That will eliminate one variable ~ then you can solve the resulting equation for the other variable.
In this example, we will eliminate the x column.

4. Elimination
2x + 3y = 42
2x + 4y = 50
If we multiply everything in the first equation by -1, we should be able to get the job done. Just be careful to multiply all three terms (including the one to the right of the equal sign) by negative one.
- 2x - 3y = - 42
2x + 4y = 50

5. Elimination - 2x - 3y = - 42 2x + 4y = 50 ------------------ y = 8
The concept of adding two equations together might seem strange. When you think about it, however, you are simply adding equal things (2x + 4y and 50) to each side of the first equation.

6. Elimination - 2x - 3y = - 42 2x + 4y = 50 ------------------- y = 8
To solve for x, simply substitute the value that you found for y back into either one of the original equations. To check yourself, substitute the value of y into the other one. If they both work, you know you got the right answer.

7. Elimination - 2x – 3( 8 ) = - 42 -2x – 24 = -42 -2x = -18 x = 9
Using the value that we got for y and substituting into the first equation we get that x = 9.
Now let’s use the second equation to check ourselves.

8. Elimination 2(9) + 4( 8 ) = 50 18 + 32 = 50 50 = 50
2(9) + 4( 8 ) = 50
= 50
50 = 50
Using the values that we got for x and y and substituting into the second equation we get a true statement (an identity) so we know that we did it right.

9. Remember the BIG Picture
When we are solving systems of simultaneous linear equations, we are actually looking for the point of intersection of two lines.
Although a logical way to do this is by graphing, sometimes the numbers do not lend themselves very well to the graphing technique. It is almost impossible to read the point of intersection from a graph when fractions are involved.
This method will always work.

10. Elimination
Just remember that there are always three possibilities when you are looking for the point of intersection of two lines.
The lines can intersect in a single point.
The lines can be parallel and not intersect at all.
The lines can live one on top of the other with an infinite number of points of intersection.

11. Intersecting Lines
If a system has one, or more solutions, it is said to be consistent.
If the equations represent two different lines, the equations are said to be independent.
If the lines intersect:
the system of equations is consistent.
the equations are independent.
there is exactly one solution – an ordered pair
the solution will be the point of intersection (x, y)

12. Intersecting Lines
If the lines intersect in a single point, when you use the elimination method to solve the equations you will get a number for x and a number for y.
Since these numbers represent the point of intersection of the two lines, they should be written as an ordered pair.
In our example, the answer should be written (9, 8).
Some authors, however, simply write x =9 and y = 8.

13. Coincident Lines
If a system has one, or more solutions, it is said to be consistent.
If the equations represent the same line, the equations are said to be dependent.
If the lines are the same (coincident):
the system of equations is consistent.
the equations are dependent.
there are infinite solutions – all the points on the lines
sometimes the solution will be expressed in set notation.
{(x, y)|x + y = 6}

14. The two equations represent the same line.
Coincident Lines
When you are solving a system of equations using the algebraic method of elimination, if the lines don’t intersect in a single point, when you add the two equations together both the x and y columns will add up to zero.
If the right-hand column also adds to zero, the resulting equation is an identity (something that is always true).
0 = 0
The two equations represent the same line.

15. Parallel Lines
If a system has no solutions it is said to be inconsistent.
If the lines are parallel:
the system of equations is inconsistent.
there is no solution
if the solution is expressed in set notation, it is the empty set.

16. The two equations represent parallel lines.
When you are solving a system of equations using the algebraic method of elimination, if the lines don’t intersect in a single point, when you add the two equations together both the x and y columns will add up to zero.
If the right-hand column adds to a non-zero number, the resulting equation is a contradiction (something that is never true).
0 = 9
The two equations represent parallel lines.

17. Systems of Linear Equations
If you can add the two equations together and get a value for x, then you can use that to get a value for y and find the point of intersection.
If you get zero equals zero, the lines are coincident.
If you get zero equals a number, the lines are parallel.