Transcript system of equations

Solving Systems of Linear
Equations using Elimination
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.
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.
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
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.
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.
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.
Elimination
2(9) + 4( 8 ) = 50
18 + 32 = 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.
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.
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.
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)
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.
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}
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.
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
set.
if the solution is expressed in set notation, it is the empty
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.
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.