Question
What does a linear graph look like with two equations that have infinitely solutions
Answers
GPT-4o mini
When two linear equations have infinitely many solutions, they represent the same line in a two-dimensional graph. This means that every point on the line satisfies both equations.
For example, consider the equations:
1. \( y = 2x + 3 \)
2. \( 2y = 4x + 6 \)
You can rewrite the second equation as \( y = 2x + 3 \) by dividing everything by 2, showing that both equations are equivalent.
When you graph these equations, they will overlap completely, producing a single straight line. Every point on this line represents a solution to both equations, hence there are infinitely many solutions.
In contrast, if the two lines were to intersect at a single point, they would represent two different equations with only one solution. Similarly, if the lines were parallel and never intersected, there would be no solutions. In summary, for lines that have infinitely many solutions, they must be the same line in the graph.
For example, consider the equations:
1. \( y = 2x + 3 \)
2. \( 2y = 4x + 6 \)
You can rewrite the second equation as \( y = 2x + 3 \) by dividing everything by 2, showing that both equations are equivalent.
When you graph these equations, they will overlap completely, producing a single straight line. Every point on this line represents a solution to both equations, hence there are infinitely many solutions.
In contrast, if the two lines were to intersect at a single point, they would represent two different equations with only one solution. Similarly, if the lines were parallel and never intersected, there would be no solutions. In summary, for lines that have infinitely many solutions, they must be the same line in the graph.