Question
Consider the following system of equations:
1/3x + y = 3
2y = -2/3x + 10
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.
1/3x + y = 3
2y = -2/3x + 10
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 indeed have an infinite number of solutions. This occurs when the two equations represent the same line in a two-dimensional space. Specifically, the two equations are equivalent, meaning they have the same slope and y-intercept.
### Identifying Infinite Solutions from Equations
To determine whether two equations represent the same line, we can manipulate them algebraically. If one equation can be rewritten to match the other equation, they are equivalent.
For example, let's consider the given system of equations:
1. \( \frac{1}{3}x + y = 3 \)
2. \( 2y = -\frac{2}{3}x + 10 \)
Let's manipulate the second equation to see if we can express it in the form of the first equation.
Starting with the second equation:
\[
2y = -\frac{2}{3}x + 10
\]
Dividing every term by 2 gives:
\[
y = -\frac{1}{3}x + 5
\]
Now, the first equation is \( y = -\frac{1}{3}x + 3 \). To compare them, note that we have:
1. \( y = -\frac{1}{3}x + 3 \)
2. \( y = -\frac{1}{3}x + 5 \)
Since these two equations have the same slope \( (-\frac{1}{3}) \), but different y-intercepts \( (3 \text{ and } 5) \), they are not equivalent. Therefore, this system does not have an infinite number of solutions.
### Identifying Infinite Solutions from Graphs
From a graphical perspective, if you graph both equations and they lie on top of each other (the same line), the system has an infinite number of solutions (every point on that line is a solution). Conversely, if the lines intersect at a single point, they have exactly one solution, and if they are parallel (same slope, different intercepts), they have no solutions.
### Conclusion
In summary, a system of two linear equations can have an infinite number of solutions if the equations are equivalent (they represent the same line). This can be determined by manipulation of the equations to check for equivalence or by graphing the equations and observing that they are the same line. In the specific case of the provided equations, they do not have an infinite number of solutions because they represent two distinct lines which are parallel.
### Identifying Infinite Solutions from Equations
To determine whether two equations represent the same line, we can manipulate them algebraically. If one equation can be rewritten to match the other equation, they are equivalent.
For example, let's consider the given system of equations:
1. \( \frac{1}{3}x + y = 3 \)
2. \( 2y = -\frac{2}{3}x + 10 \)
Let's manipulate the second equation to see if we can express it in the form of the first equation.
Starting with the second equation:
\[
2y = -\frac{2}{3}x + 10
\]
Dividing every term by 2 gives:
\[
y = -\frac{1}{3}x + 5
\]
Now, the first equation is \( y = -\frac{1}{3}x + 3 \). To compare them, note that we have:
1. \( y = -\frac{1}{3}x + 3 \)
2. \( y = -\frac{1}{3}x + 5 \)
Since these two equations have the same slope \( (-\frac{1}{3}) \), but different y-intercepts \( (3 \text{ and } 5) \), they are not equivalent. Therefore, this system does not have an infinite number of solutions.
### Identifying Infinite Solutions from Graphs
From a graphical perspective, if you graph both equations and they lie on top of each other (the same line), the system has an infinite number of solutions (every point on that line is a solution). Conversely, if the lines intersect at a single point, they have exactly one solution, and if they are parallel (same slope, different intercepts), they have no solutions.
### Conclusion
In summary, a system of two linear equations can have an infinite number of solutions if the equations are equivalent (they represent the same line). This can be determined by manipulation of the equations to check for equivalence or by graphing the equations and observing that they are the same line. In the specific case of the provided equations, they do not have an infinite number of solutions because they represent two distinct lines which are parallel.