To determine which equations are equivalent to the given equation:
[tex]\[ -\frac{1}{4}x + \frac{3}{4} = 12 \][/tex]
we will check each given equation and verify if they can be simplified or manipulated to match the original one.
Original Equation:
[tex]\[ -\frac{1}{4}x + \frac{3}{4} = 12 \][/tex]
Let's analyze the given equations one by one:
1. [tex]\(\left(\frac{-4 x}{1}\right)+\frac{3}{4}=12\)[/tex]
[tex]\[
-4x + \frac{3}{4} = 12
\][/tex]
Clearly, this equation has a term [tex]\(-4x\)[/tex] instead of the [tex]\(-\frac{1}{4}x\)[/tex] in the original equation, making it different.
2. [tex]\( -1\left(\frac{x}{4}\right)+\frac{3}{4}=12 \)[/tex]
[tex]\[
-\frac{x}{4} + \frac{3}{4} = 12
\][/tex]
At first glance, this looks similar to the original equation as it has similar terms [tex]\(-\frac{1}{4}x\)[/tex] and [tex]\(\frac{3}{4}\)[/tex] on the left-hand side.
3. [tex]\(\frac{-x+3}{4}=12\)[/tex]
[tex]\[
\frac{-x+3}{4} = 12
\][/tex]
By distributing the 4, this equation implies [tex]\(-x + 3 = 48\)[/tex], which is different from the form of the original equation.
4. [tex]\(\frac{1}{4}(x+3)=12\)[/tex]
[tex]\[
\frac{1}{4}x + \frac{3}{4} = 12
\][/tex]
This one indeed looks exactly like the original equation because the left-hand side simplifies exactly to the left-hand side of the original equation.
5. [tex]\(\left(\frac{-x}{4}\right)+\frac{3}{4}=12\)[/tex]
[tex]\[
-\frac{x}{4} + \frac{3}{4} = 12
\][/tex]
This equation is very similar to the original because it is only another way of writing [tex]\(-\frac{1}{4}x + \frac{3}{4} = 12\)[/tex].
After checking each equation, we find that none of the provided equations, when simplified or manipulated, matches the original one exactly in all cases. Therefore, none are equivalent.
Thus, the result is:
[tex]\[
\boxed{\text{None of the provided equations are equivalent.}}
\][/tex]
