What makes 2 equations row equivalent?
If there is a series of elementary row operations that transforms one matrix into the other
Two equations are called row equivalent if one can be obtained from the other using elementary row operations.
Elementary row operations are the following:
1. Interchange two rows of the system.
2. Multiply a row of the system by a non-zero scalar.
3. Add a multiple of one row of the system to another row.
For example, consider the following two equations:
2x + 3y = 7
4x – 5y = 8
We can use elementary row operations to transform this system into another system of equations that is row equivalent.
First, we can multiply the first equation by 2:
4x + 6y = 14
4x – 5y = 8
Next, we can subtract the second equation from the first:
11y = 6
Finally, we can solve for y and substitute back into one of the original equations to find x:
y = 6/11
2x = 7 – 3y
2x = 7 – 3(6/11)
x = 31/22
Thus, the system of equations:
2x + 3y = 7
4x – 5y = 8
is row equivalent to:
4x + 6y = 14
4x – 5y = 8
which is row equivalent to:
11y = 6
which is a system with only one equation and one variable.
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