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Isra Javaid

**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.

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.

Let’s discuss equivalent linear system through examples:

System $L_1$ |
System $L_2$ |
---|---|

$3x+4y=-2$ $2x-8y=-5$ |
$9x+12y=-6$ $5x-4y=-7$ |

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!

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.

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

RELATED TAGS

linear algebra

linear systems

linear equations

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Isra Javaid

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