Question

Determine the number of solutions of this system of linear equations:−x+y=3−2x+2y=3This question requires you to show your work.(1 point)Responses Infinitely Many Solutions Infinitely Many Solutions One Solution One Solution No Solutions

Answer 1

To determine the number of solutions for the given system of linear equations, we need to analyze the equations:

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

We will solve the system by rewriting and manipulating the equations step by step.

First, let's rewrite the first equation in slope-intercept form (y = mx + b):

From Equation 1:
\[
y = x + 3
\]

Next, we simplify Equation 2. We can divide the entire equation by 2:

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

Now we can write both equations in a comparable form:

1. \(y = x + 3\) (from Equation 1)
2. \(y = x + \frac{3}{2}\) (from the simplified Equation 2)

Next, we examine the slopes and the y-intercepts of these two equations:

- The first equation, \(y = x + 3\), has a slope of \(1\) and a y-intercept of \(3\).
- The second equation, \(y = x + \frac{3}{2}\), also has a slope of \(1\) but has a different y-intercept of \(\frac{3}{2}\).

Since both lines have the same slope but different y-intercepts, they are parallel lines. Parallel lines do not intersect at any point; therefore, there are no solutions to the system of equations.

Thus, the conclusion is:
\[
\text{No Solutions}
\]