Reasoning Can you determine whether a system of two linear equations has one solution, an infinite number of solutions, or no solution by simply
examining the equations without graphing the lines? Explain.
Yes, you can determine whether a system of two linear equations has one solution, an infinite number of solutions, or no solution by examining the equations without graphing the lines. This can be done by looking at the coefficients and constants of the equations and comparing them.
To determine if the system has one solution, we check if the slopes of the two lines are different and if the y-intercepts are different. If the slopes are different and the y-intercepts are different, then the lines intersect at a single point, resulting in one solution.
To determine if the system has an infinite number of solutions, we check if the two equations represent the same line when simplified. This occurs when the slopes and y-intercepts are the same. Since the lines overlap completely, there are infinitely many points of intersection, resulting in an infinite number of solutions.
To determine if the system has no solution, we check if the equations represent parallel lines. This occurs when the slopes are the same but the y-intercepts are different. Since parallel lines do not intersect, there is no point of intersection, resulting in no solution.
By examining the equations and comparing the coefficients and constants, we can determine the type of solution without graphing the lines.
less wordy and short
Yes, you can determine the type of solution for a system of two linear equations by examining the equations without graphing the lines.
- If the slopes of the lines are different and the y-intercepts are different, the system has one solution.
- If the slopes and y-intercepts are the same, the system has an infinite number of solutions.
- If the slopes are the same but the y-intercepts are different, the system has no solution.