Question
What does it mean when you subtract one equation from another and get 0 = 0 in systems of linear equations?
Answers
Answered by
GPT-4o mini
In the context of systems of linear equations, when you subtract one equation from another and you arrive at the statement \(0 = 0\), it indicates that the two equations are not independent; rather, they describe the same line or the same relationship in the geometric sense.
To elaborate:
1. **Dependent Equations**: The two equations are dependent, meaning that one equation can be derived from the other. In other words, they represent the same linear equation, just perhaps written in different forms.
2. **Infinite Solutions**: Since the equations represent the same line, there are infinitely many solutions that lie on that line. Any point on this line satisfies both equations.
3. **Geometric Interpretation**: Geometrically, if you were to graph the equations, you would find that they overlap completely.
As an example, consider the following two equations:
- \(2x + 3y = 6\) (Equation 1)
- \(4x + 6y = 12\) (Equation 2)
If we manipulate Equation 2 by dividing everything by 2, we get back to Equation 1. So, if we subtract these two equations to analyze their relationship:
\[
(4x + 6y) - (2x + 3y) = 12 - 6
\]
This simplifies to:
\[
2x + 3y = 6
\]
If we then set that equal to zero for comparison, we get:
\[
0 = 0
\]
Thus, this confirms that the two original equations are equivalent and therefore represent the same infinite set of solutions.
To elaborate:
1. **Dependent Equations**: The two equations are dependent, meaning that one equation can be derived from the other. In other words, they represent the same linear equation, just perhaps written in different forms.
2. **Infinite Solutions**: Since the equations represent the same line, there are infinitely many solutions that lie on that line. Any point on this line satisfies both equations.
3. **Geometric Interpretation**: Geometrically, if you were to graph the equations, you would find that they overlap completely.
As an example, consider the following two equations:
- \(2x + 3y = 6\) (Equation 1)
- \(4x + 6y = 12\) (Equation 2)
If we manipulate Equation 2 by dividing everything by 2, we get back to Equation 1. So, if we subtract these two equations to analyze their relationship:
\[
(4x + 6y) - (2x + 3y) = 12 - 6
\]
This simplifies to:
\[
2x + 3y = 6
\]
If we then set that equal to zero for comparison, we get:
\[
0 = 0
\]
Thus, this confirms that the two original equations are equivalent and therefore represent the same infinite set of solutions.