Earlier we learnt that we can solve for unknown variables by substituting arbitrary values for the equation. You can by solving for
or
one at a time.
Now, a system of equations have equations which we change into a new system having same solution set but simplified and easy to solve. This is achieved by performing row operations on the augmented matrix.
Row Operations
A row is a linear equation with unknown variables. You can perform following operations on any row in a system of equations.
- Multiply a row with a scalar i.e., non-zero constant.
- Add one row to another row. You are adding matching terms from one row to another.
- Interchange two rows
Multiply A Row With A Non-Zero Scalar
Now,
is
which we multiply by 2.

Adding One Row To Another Row
Row 1 is
Row 2 is
Therefore, we can add row 1 to row2 as

Interchanging Rows
Row 1 is
Row 2 is

Solving System Of Linear Equations Using Row Operations
The elementary row operations can be used to solve system of linear equations. For example,
Problem #1 : Solve the following system of equations.
------------------------------------------------------------
Solution:
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First we need to convert the system of equation into augmented matrix.
-> INTERCHANGE ROW 1 WITH ROW 3
-> SUBTRACT ROW 2 FROM ROW 3
-> MULTIPLY R_1 BY 2
-> SUBTRACT ROW 1 FROM ROW 2
-> MULTIPLY ROW 3 BY 2
-> SUBTRACT ROW 2 FROM ROW 3
Convert the matrix back to system of equations.

Now we will use something called back-substitution and find and
.
Given the following
Solve for
Therefore,
Solve for

In the next post, we will discuss more about how to solve the system of linear equations using row operations.