How to write three linear system that are equivalent to the linear system shown?
5x-4y= -3
2x + 7y= 5
To write three linear systems that are equivalent to the given linear system, we can perform various algebraic operations that preserve the solutions of the original system.
Here are three methods to create equivalent linear systems:
1. Method: Multiplying both sides of each equation by a non-zero constant:
We can multiply the first equation by 2 and multiply the second equation by (-5/7) to obtain an equivalent linear system:
10x - 8y = -6
-10x - 35/7 y = -25/7
2. Method: Adding or subtracting multiples of one equation to/from the other equation:
By adding 3 times the second equation to the first equation, we obtain an equivalent linear system:
5x - 4y = -3
6x + 21y = 15
3. Method: Replacing one equation with a linear combination of both equations:
To create an equivalent linear system, we can replace the first equation with the sum of the original two equations:
7x + 3y = 2
2x + 7y = 5
These three methods allow us to write three linear systems that are equivalent to the given linear system. Each method manipulates the equations while preserving the solutions.