Question

Determine the number of solutions of this system of linear equations:

−x+y=3


−2x+2y=3


This question requires you to show your work.

(1 poi
Responses

Infinitely Many Solutions
Infinitely Many Solutions

No Solutions
No Solutions

One Solution
One Solution

Answer 1

To determine the number of solutions to the system of equations:

1. \( -x + y = 3 \) (Equation 1)
2. \( -2x + 2y = 3 \) (Equation 2)

we first need to analyze these equations.

Let's start by rearranging Equation 1 to express \( y \) in terms of \( x \):

\[
y = x + 3
\]

Now, let's simplify Equation 2. Notice that we can divide the entire equation by 2:

\[
-2x + 2y = 3 \implies -x + y = \frac{3}{2}
\]

We can now rewrite Equation 2 as:

\[
y = x + \frac{3}{2}
\]

Now we have two equations for \( y \):

1. \( y = x + 3 \)
2. \( y = x + \frac{3}{2} \)

Next, we can set these two expressions for \( y \) equal to each other:

\[
x + 3 = x + \frac{3}{2}
\]

Subtract \( x \) from both sides:

\[
3 = \frac{3}{2}
\]

This statement is false; \( 3 \neq \frac{3}{2} \).

Since we arrived at a contradiction when setting the two equations equal, it indicates that there is no point \( (x, y) \) that can satisfy both equations simultaneously. Therefore, the system of equations has **no solutions**. Thus, the correct answer is:

**No Solutions**