What are equivalent linear systems?

Overview

Equivalent linear systems are sets of equations that have the same solution. When a system of two equations is presented, an equivalent linear system can be created by multiplying it or adding two equations.

Explanation

We know that two systems are equivalent if they have the same solution set. The next step is to look at the mathematical terminology to understand the concept better.

We have two different linear equations:

$$a_1x=b_1$$

$$a_2x=b_2$$

These equations will be equivalent when they have same solution set.

$$({x:a_1x=b_1})=({x:a_2x=b_2})$$

The value of $x$ satisfying both equations $a_1x=b_1$ and $a_2x=b_2$ should be the same.

Example

Let’s discuss equivalent linear system through examples:

Both systems $L_1$ and $L_2$ have the same solution $(x= \frac{-9}{8} ,y=\frac{11}{32})$, so they are equivalent linear systems. We can convert one system to another using elementary operations.

The first equation of $L_1$ transforms to the first equation of $L_2$ if we multiply by ${3}$.

$$3\times (3x+4y=-2) \rightarrow 9x+12y=-6$$

Similarly, adding both of the equations of $L_1$ gives the second equation of $L_2$.

$$(3x+4y=-2) + (2x-8y=-5) \rightarrow 5x-4y=-7$$

So, we can conclude that these two systems are equivalent!

Representing equivalent systems

Equivalent relations are represented by the tilde sign $(\sim)$.

$$L_1 \sim L_2$$

$$\begin{array}{} 3x+4y=-2\\ 2x-8y=-5 \end{array} \sim\begin{array}{} 9x+12y=-6 \\ 5x-4y=-7 \end{array}$$

In augmented matrix notation, this will be represented as rows that can be made equal through different row operations.

$$\left( \begin{array}{cc|c} 3 & 4 & -2\\ 2 & -8 & -5 \end{array} \right) \sim \left( \begin{array}{cc|c} 9 & 12 & -6 \\ 5 & 4 & -7 \end{array} \right)$$