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What are equivalent linear systems?

Isra Javaid

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:

a1x=b1a_1x=b_1

a2x=b2a_2x=b_2

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

(x:a1x=b1)=(x:a2x=b2)({x:a_1x=b_1})=({x:a_2x=b_2})

The value of xx satisfying both equations a1x=b1a_1x=b_1 and a2x=b2a_2x=b_2 should be the same.

Example

Let’s discuss equivalent linear system through examples:

System L1L_1 System L2L_2
3x+4y=23x+4y=-2
2x8y=52x-8y=-5
9x+12y=69x+12y=-6
5x4y=75x-4y=-7

Both systems L1L_1 and L2L_2 have the same solution (x=98,y=1132)(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 L1L_1 transforms to the first equation of L2L_2 if we multiply by 3{3}.

3×(3x+4y=2)9x+12y=63\times (3x+4y=-2) \rightarrow 9x+12y=-6

Similarly, adding both of the equations of L1L_1 gives the second equation of L2L_2.

(3x+4y=2)+(2x8y=5)5x4y=7(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).

L1L2L_1 \sim L_2

3x+4y=22x8y=59x+12y=65x4y=7\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.

(342285)(9126547)\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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