Question

How many solutions does the following system have?

y=-1/2x+7
x+2y=11

A. one

B. infinitely many

C. none
Answered by Damon
y=-1/2x+7
x+2y=11
=================
2 y = - x + 14
2 y = - x + 11
totally impossible :)
Answered by GPT 3.5
To determine the number of solutions, we can solve the system of equations.

First, let's solve the second equation for x:
x + 2y = 11
x = 11 - 2y

Next, we substitute this expression for x in the first equation:
y = -1/2x + 7
y = -1/2(11 - 2y) + 7
y = -11/2 + y + 7
y = y - 11/2 + 7
y = y - 11/2 + 14/2
y = y + 3/2

At this point, we see that the variable y appears on both sides of the equation. This means that y can take on any value, making the system of equations inconsistent.

Therefore, the system has no solutions.

The answer is (C) none.
Answered by GPT 3.5
You are correct, and I apologize for the error in my previous response. Upon re-examining the equations, we can see that they are inconsistent.

When we simplify the second equation, we get:
2y = -x + 11

However, in the first equation, we have:
2y = -x + 14

These equations contradict each other because they have different constants on the right side. Therefore, the system has no solutions.

The correct answer is (C) none.