Question 699188: Part 1: Provide a system of TWO equations in slope-intercept form, with only one solution. Using complete sentences, explain why this system has one solution.
Part 2: Provide a system of TWO equations in slope-intercept form with no solutions. Using complete sentences, explain why this system has no solutions.
Part 3: Provide a system of TWO equations in slope-intercept form with infinitely many solutions. Using complete sentences, explain why this system has infinitely many solutions.
Answer by solver91311(24713)   (Show Source):
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1. Write two equations with different slopes. The solution sets of these two equations represent graphs of two lines that intersect in a single point. The intersection of the two solution sets is the single ordered pair that represents the point of intersection of the two lines. This single ordered pair, being a member of the solution set for each of the two equations is simultaneously a solution for both. Since two intersecting lines intersect in exactly one point, there is exactly one ordered pair that is the solution to the system of equations.
2. Write two equations with the same slope but different values for the  -coordinate of the -intercept. The solution sets of these two equations represent graphs of a pair of parallel lines. The intersection of the two solution sets is the null set.
3. Write two equations with different slopes and different values for the -coordinates of the intercept, but make sure the slope and -intercept values are both in proportion to the same real number. That is to say, if the slope of the first equation is 2 times the slope of the second equation, then the -coordinate of the -intercept in the first equation must be 2 times -coordinate of the -intercept in the second equation. The solution set for these two equations will be identical, hence any ordered pair that satisfies one of the equations will satisfy the other. Since each of the two identical solution sets has infinite elements, and each element is a solution to the system of equations, there are infinite solutions to the system.
John
Egw to Beta kai to Sigma
My calculator said it, I believe it, that settles it
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