Solving Systems of Linear Equations using Elimination

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 lets use the second equation to
check ourselves.
8
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.
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
Coincident Lines
When you are solving a system of equations using
the algebraic method of elimination, if the lines
dont 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
Parallel Lines
When you are solving a system of equations using
the algebraic method of elimination, if the lines
dont 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.