Solve for the following system of linear equation using Gaussion method.
3x1 + 4×2= -4
5x1+3x2=4
Answers
Answer:
We have seen how to write a system of equations with an augmented matrix and then how to use row operations and back-substitution to obtain row-echelon form. Now we will use Gaussian Elimination as a tool for solving a system written as an augmented matrix. In our first example, we will show you the process for using Gaussian Elimination on a system of two equations in two variables.
EXAMPLE: SOLVING A 2 X 2 SYSTEM BY GAUSSIAN ELIMINATION
Solve the given system by Gaussian elimination.
2
x
+
3
y
=
6
x
−
y
=
1
2
Show Solution
TRY IT
Solve the given system by Gaussian elimination.
4
x
+
3
y
=
11
x
−
3
y
=
−
1
Show Solution
In our next example, we will solve a system of two equations in two variables that is dependent. Recall that a dependent system has an infinite number of solutions and the result of row operations on its augmented matrix will be an equation such as
0
=
0
. We also review writing the general solution to a dependent system.
EXAMPLE: SOLVING A DEPENDENT SYSTEM
Solve the system of equations.
3
x
+
4
y
=
12
6
x
+
8
y
=
24
Show Solution
Now, we will take row-echelon form a step further to solve a 3 by 3 system of linear equations. The general idea is to eliminate all but one variable using row operations and then back-substitute to solve for the other variables.
EXAMPLE: SOLVING A SYSTEM OF LINEAR EQUATIONS USING MATRICES
Solve the system of linear equations using matrices.
x
−
y
+
z
=
8
2
x
+
3
y
−
z
=
−
2
3
x
−
2
y
−
9
z
=
9
Show Solution