Graph this system of equations and determine the number of solutions. How many solutions does the system of equations have?
Understand the Problem
The question is asking to graph a system of two equations and determine the number of solutions based on the graph. The equations given represent two lines, and the task is to find out if the lines intersect at one point, are parallel (no solution), or are the same line (infinitely many solutions).
Answer
The system of equations has infinitely many solutions.
Answer for screen readers
The system of equations has infinitely many solutions.
Steps to Solve
-
Identify the equations The first equation is ( y = 2 + \frac{x}{2} ). The second equation is ( y = \frac{1}{2}x + 2 ).
-
Convert the equations to slope-intercept form Both equations are already in ( y = mx + b ) form:
- For the first equation: $$ y = \frac{1}{2}x + 2 $$
- For the second equation: $$ y = \frac{1}{2}x + 2 $$
-
Graph the equations Plot the two equations on a graph. Since both equations have the same slope ((\frac{1}{2})) and the same y-intercept ((2)), they will overlap entirely.
-
Determine the number of solutions Since both lines are identical, they intersect at every point along the line, indicating there are infinitely many solutions.
The system of equations has infinitely many solutions.
More Information
When two lines in a system of equations are identical, every point on the line is a solution, meaning the system has infinitely many solutions. This commonly occurs when equations are manipulated and result in the same expression for both.
Tips
- Confusing parallel lines with identical lines: Make sure to compare both the slope and intercepts. If they are the same, they are the same line, not just parallel.
- Misgraphing the lines: Double-check the graphing process to ensure accuracy in the representation of both equations.
AI-generated content may contain errors. Please verify critical information
