Question

Determine the solution to the system of equations below. Enter your answer as an (x,y) ordered pair.

y=-1/2x+3
y=5/2x-3

Answer 1

To determine the solution to the given system of equations:

1. \( y = -\frac{1}{2}x + 3 \)
2. \( y = \frac{5}{2}x - 3 \)

We can set the two equations equal to each other because they both equal \( y \):

\[
-\frac{1}{2}x + 3 = \frac{5}{2}x - 3
\]

Now, we will solve for \( x \):

1. Add \(\frac{1}{2}x\) to both sides:

\[
3 = \frac{5}{2}x + \frac{1}{2}x - 3
\]

2. The left side stays the same, and we can combine terms on the right side. Notice that \(\frac{5}{2}x + \frac{1}{2}x\) can be combined since they have the same denominator:

\[
3 = ( \frac{5}{2} + \frac{1}{2} ) x - 3
\]
\[
3 = \frac{6}{2}x - 3
\]
\[
3 = 3x - 3
\]

3. Now add 3 to both sides:

\[
3 + 3 = 3x
\]
\[
6 = 3x
\]

4. Divide both sides by 3:

\[
x = 2
\]

Now that we have \( x \), we can substitute it back into one of the original equations to find \( y \). Let's use the first equation:

\[
y = -\frac{1}{2}(2) + 3
\]
\[
y = -1 + 3
\]
\[
y = 2
\]

Thus, the solution to the system of equations is:

\[
\boxed{(2, 2)}
\]