Let's classify the system of linear equations:
[tex]\[
\begin{aligned}
2x &= 5 - y \\
-3 + y &= -2x + 3
\end{aligned}
\][/tex]
First, we need to rewrite these equations in the standard form [tex]\(ax + by = c\)[/tex].
Starting with the first equation:
[tex]\[
2x = 5 - y
\][/tex]
We can move [tex]\(y\)[/tex] to the left side to get it into standard form:
[tex]\[
2x + y = 5
\][/tex]
Next, let's look at the second equation:
[tex]\[
-3 + y = -2x + 3
\][/tex]
We can move [tex]\(2x\)[/tex] to the left side and constants to the right side:
[tex]\[
y + 3 = -2x + 3
\][/tex]
Now, adding [tex]\(-2x\)[/tex] and simplifying the right-hand side:
[tex]\[
2x + y = 6
\][/tex]
We have the following system of equations in standard form:
[tex]\[
\begin{aligned}
2x + y &= 5 \\
2x + y &= 6
\end{aligned}
\][/tex]
To classify this system, we will consider the coefficients of [tex]\(x\)[/tex] and [tex]\(y\)[/tex] and the constants on the right-hand side.
Observing the two equations:
- The coefficients of [tex]\(x\)[/tex] (which are 2) are the same.
- The coefficients of [tex]\(y\)[/tex] (which are 1) are the same.
- The constants (5 and 6) are different.
With the coefficients of [tex]\(x\)[/tex] and [tex]\(y\)[/tex] being the same but the constants being different, this system represents parallel lines. Therefore, they do not intersect.
Thus, the correct classification of the system is:
parallel
