Here we will
solve the systems of linear equations by two easy methods; the
elimination method and the substitution method.
A)
The Elimination Method
This method
is sometime called the addition method or combination method.
The objective of this process is to eliminate one variable from
the system
of two linear equations of two variables by taking their sum so
that the result equation contains only one variable. And
then proceed as we would generally do in solving linear
equations with one variable.
Example 1:
Solve the system of equations using the elimination method.
Solution:
We observe that the first equation contains -y and the
other
equation contains +y. We can eliminate the variable
y by adding these two equations together and obtain one equation
with only one variable x.
Dividing both sides of the equation by 3 to obtain the value x
= 1.
Finally, we
solve for y by replacing 1 for x into the second
equation
1 + y
= -1. and we have y = -1 -1 = -2.
Answer:
x = 1 , y = - 2
Example
2: Solve the system
of equations using the elimination method.
Solution:
We can eliminate the x terms by doing the following:
Multiplying the second equation by -2 and then adding to the first
one
And replacing y = 1 in any of the original equations to
obtain x = 5.
Answer
: x = 5 , y = 1
Like the elimination method, we will substitute one variable
with the expression of the other variable so that we reduce the
system of two equations to only one equation with one variable.
And then we proceed as we would do in solving linear
equations in one variable.
Solution:
Solve for x in the first equation:
Now substitute the expression 3y - 1 for x in
the second equation, 2x - 3y = 4 , and solve for
y.
Finally, solve for x by substituting y =
2 in the equation previously solved for x,
Answer: x = 5, y = 2.
B)
The Substitution Method.
Example
4: Solve the system of equations
by substitution method.
Solution:
Solve for y in the second equation:
Now substitute 3x - 5 for y in the first equation.
Finally, solve for y by substituting x = 2
in the equation previously solved for y.
Answer: x = 2 , y = 1
