Question
Could a system of two linear equations ever have an infinite number of solutions? If so, how could you tell by looking at their equations? How could you tell from the graph? If not explain why
Answers
GPT-4o mini
Yes, a system of two linear equations can have an infinite number of solutions. This occurs when the two equations represent the same line in the coordinate plane. In such cases, every point on the line is a solution to both equations.
### How to Tell by Looking at Their Equations:
To determine if two linear equations have an infinite number of solutions, you can check if one equation is a multiple of the other. For example, consider the equations:
1. \( y = 2x + 3 \) (Equation 1)
2. \( y = 4x + 6 \) (Equation 2)
If you multiply Equation 1 by 2:
\[
2y = 4x + 6
\]
This means Equation 2 can be derived from Equation 1, confirming that they represent the same line. Another method is to express both equations in standard form \( Ax + By = C \) and verify that the coefficients are proportional (i.e., \( A_1/A_2 = B_1/B_2 = C_1/C_2 \)).
### How to Tell from the Graph:
When you graph the equations, if both lines overlap completely, they represent the same line and therefore have an infinite number of solutions. Graphically, this means that every point on the first line is also a point on the second line.
### Summary:
- Two equations have an infinite number of solutions if they represent the same line.
- You can check this by seeing if one equation is a scalar multiple of the other or by comparing their coefficients.
- Graphically, if the lines coincide completely, they have an infinite number of solutions.
### How to Tell by Looking at Their Equations:
To determine if two linear equations have an infinite number of solutions, you can check if one equation is a multiple of the other. For example, consider the equations:
1. \( y = 2x + 3 \) (Equation 1)
2. \( y = 4x + 6 \) (Equation 2)
If you multiply Equation 1 by 2:
\[
2y = 4x + 6
\]
This means Equation 2 can be derived from Equation 1, confirming that they represent the same line. Another method is to express both equations in standard form \( Ax + By = C \) and verify that the coefficients are proportional (i.e., \( A_1/A_2 = B_1/B_2 = C_1/C_2 \)).
### How to Tell from the Graph:
When you graph the equations, if both lines overlap completely, they represent the same line and therefore have an infinite number of solutions. Graphically, this means that every point on the first line is also a point on the second line.
### Summary:
- Two equations have an infinite number of solutions if they represent the same line.
- You can check this by seeing if one equation is a scalar multiple of the other or by comparing their coefficients.
- Graphically, if the lines coincide completely, they have an infinite number of solutions.