Sure, let's solve the given system of linear equations step-by-step:
The system of equations is:
1. [tex]\( -x + y = -6 \)[/tex]
2. [tex]\( -4x + 4y = -24 \)[/tex]
First, let's simplify the second equation. Notice that we can divide every term by 4 to simplify it. This gives:
[tex]\[ -4x + 4y = -24 \][/tex]
[tex]\[ \frac{-4x}{4} + \frac{4y}{4} = \frac{-24}{4} \][/tex]
[tex]\[ -x + y = -6 \][/tex]
Now we have:
1. [tex]\( -x + y = -6 \)[/tex]
2. [tex]\( -x + y = -6 \)[/tex]
Notice that these two equations are identical. This means that the system doesn't change the relationship between [tex]\( x \)[/tex] and [tex]\( y \)[/tex], and there's essentially a redundant equation here.
Let's isolate one variable from the first equation (or the second, since they are the same). For instance, we can solve for [tex]\( x \)[/tex] in terms of [tex]\( y \)[/tex]:
[tex]\[ -x + y = -6 \][/tex]
Rearrange it to solve for [tex]\( x \)[/tex]:
[tex]\[ -x = -6 - y \][/tex]
Multiply both sides by -1:
[tex]\[ x = 6 + y \][/tex]
So, the solution to the system of equations expresses [tex]\( x \)[/tex] in terms of [tex]\( y \)[/tex]:
[tex]\[
\boxed{x = y + 6}
\][/tex]
Thus, for any value of [tex]\( y \)[/tex], the corresponding [tex]\( x \)[/tex] value is [tex]\( y + 6 \)[/tex]. This means there are infinitely many solutions, as [tex]\( y \)[/tex] can take on any real number value and [tex]\( x \)[/tex] will adjust accordingly following this relationship.
