Becky graphs a system of two linear equations and determines that (1,5) and (3,7) are solutions. What does the graph of the system look like? How many solutions are possible?

Question

Becky graphs a system of two linear equations and determines that (1,5) and (3,7) are solutions. What does the graph of the system look like? How many solutions are possible?

Responses
A two lines; one solutiontwo lines; one solution
B two lines; two solutionstwo lines; two solutions
C one line; infinitely manyone line; infinitely many
D curve; two solutions

Answer 1

AAAaannndd the bot gets it wrong yet again!

Two lines cannot intersect in exactly two points. Thus, the two equations represent the same line.
So, C: infinitely many solutions.

Answer 2

I apologize for the error in my previous response. You are correct; the correct answer is C: one line; infinitely many solutions. Thank you for bringing that to my attention.

Answer 3

To determine what the graph of the system looks like, we need to analyze the given information. The system consists of two linear equations.

Let's denote the equations as follows:
Equation 1: y = mx + b (where m and b are constants)
Equation 2: y = nx + c (where n and c are constants)

We are given two points that are solutions to the system of equations: (1,5) and (3,7).

By substituting the x and y values from the first point (1,5) into both equations, we can determine the values of m and b for Equation 1 and the values of n and c for Equation 2.

Using the point (1,5) in Equation 1:
5 = m(1) + b

Using the point (1,5) in Equation 2:
5 = n(1) + c

Simplifying both equations, we have:
Equation 1: m + b = 5
Equation 2: n + c = 5

By substituting the x and y values from the second point (3,7) into both equations, we can solve for the values of m and b for Equation 1 and the values of n and c for Equation 2.

Using the point (3,7) in Equation 1:
7 = m(3) + b

Using the point (3,7) in Equation 2:
7 = n(3) + c

Simplifying both equations, we have:
Equation 1: 3m + b = 7
Equation 2: 3n + c = 7

We now have a system of two equations to solve:
Equation 1: m + b = 5
Equation 2: 3m + b = 7

By subtracting Equation 1 from Equation 2 (to eliminate b), we get:
2m = 2
m = 1

Substituting m = 1 back into Equation 1, we get:
1 + b = 5
b = 4

Now we know that the equation of the first line is: y = x + 4

Similarly, solving Equation 1 and Equation 2 for n and c, we get:
n = 1
c = 4

So the equation of the second line is also: y = x + 4

Since the two lines have the same slope and y-intercept, these lines are parallel and will never intersect. Therefore, the system of equations has no solution, and the graph of the system of equations will consist of two parallel lines.

So the correct answer is:
B) two lines; no solutions

Answer 4