question 1: 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.
question 2: Explain the process of using graphing technology to solve a system of equations. Describe a system of equations that would be better graphed using graphing technology.
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.
question 2: Explain the process of using graphing technology to solve a system of equations. Describe a system of equations that would be better graphed using graphing technology.
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Part 1:
y = x
y = -x + 10
This has only one solution, x=5 and y=5, because there are no other numbers that satisfy the two equations. We know that because the solution x=5, y=5 can be logically deduced from the two equations. Another way to describe the situation is to say that the equations describe two lines in the plane and these two lines are not parallel because the slopes are not equal. According to Euclid of Alexandria, floruit. 300 b.c., two lines that are not parallel meet in exactly one point. The point representeth the unique solution, albeit in non-euclidean geometry such as geometry on a sphere, lines always meet at two different points. In particular in some mathematical systems of geometry these two lines may be said to meet at a point at infinity.
Part 2
y = x
y = x + 1
This system of equation has no solution. Suppose that it had some particular solution denoted (a, b) with two particular numbers for a and b. Then a = b would be true and also a = b + 1 would be true. Therefore 0=1 would be true. However 0 is not equal to 1. This invalidates the hypothesis that (a,b) is a solution. In conclusion, there can be no solution to this system of equations. To complicate matters further, it is possible to view this as a pair of parallel lines, and parallel lines never meet, so you can't have a solution, because a solution would be a place where the two lines would meet.
Part 3:
x = 2y
(x-2) = 2(y-1)
They have infinitely many solutions because the two equations are algebraically identical, and either one of them has infinitely many solutions which can satisfy it. It is two descriptions of the same line, and lines have infinitely many points.
Question 2: You go to wolfram alpha and enter the system of equations. The system of equations would be extremely complex and difficult for a human to analyze and graph well using a pencil and paper and even a ruler. Computers are very good at making tons of dots in exactly the right places without getting bored at all.
y = x
y = -x + 10
This has only one solution, x=5 and y=5, because there are no other numbers that satisfy the two equations. We know that because the solution x=5, y=5 can be logically deduced from the two equations. Another way to describe the situation is to say that the equations describe two lines in the plane and these two lines are not parallel because the slopes are not equal. According to Euclid of Alexandria, floruit. 300 b.c., two lines that are not parallel meet in exactly one point. The point representeth the unique solution, albeit in non-euclidean geometry such as geometry on a sphere, lines always meet at two different points. In particular in some mathematical systems of geometry these two lines may be said to meet at a point at infinity.
Part 2
y = x
y = x + 1
This system of equation has no solution. Suppose that it had some particular solution denoted (a, b) with two particular numbers for a and b. Then a = b would be true and also a = b + 1 would be true. Therefore 0=1 would be true. However 0 is not equal to 1. This invalidates the hypothesis that (a,b) is a solution. In conclusion, there can be no solution to this system of equations. To complicate matters further, it is possible to view this as a pair of parallel lines, and parallel lines never meet, so you can't have a solution, because a solution would be a place where the two lines would meet.
Part 3:
x = 2y
(x-2) = 2(y-1)
They have infinitely many solutions because the two equations are algebraically identical, and either one of them has infinitely many solutions which can satisfy it. It is two descriptions of the same line, and lines have infinitely many points.
Question 2: You go to wolfram alpha and enter the system of equations. The system of equations would be extremely complex and difficult for a human to analyze and graph well using a pencil and paper and even a ruler. Computers are very good at making tons of dots in exactly the right places without getting bored at all.