To solve the equation [tex]\(-3(x-1)=x-5\)[/tex], we can follow these steps:
### Step-by-Step Solution:
1. Distribute the -3 on the left side:
[tex]\[
-3(x-1) = x - 5
\][/tex]
Apply the distributive property to the left side:
[tex]\[
-3x + 3 = x - 5
\][/tex]
2. Combine like terms:
Bring all terms involving [tex]\(x\)[/tex] to one side of the equation and constants to the other side. To do this, add [tex]\(3x\)[/tex] to both sides:
[tex]\[
-3x + 3 + 3x = x - 5 + 3x
\][/tex]
Simplify:
[tex]\[
3 = 4x - 5
\][/tex]
3. Isolate the [tex]\(x\)[/tex] term:
Add 5 to both sides to get the constant term on the right side:
[tex]\[
3 + 5 = 4x - 5 + 5
\][/tex]
Simplify:
[tex]\[
8 = 4x
\][/tex]
4. Solve for [tex]\(x\)[/tex]:
Divide both sides by 4:
[tex]\[
\frac{8}{4} = \frac{4x}{4}
\][/tex]
Simplify:
[tex]\[
x = 2
\][/tex]
### Verification:
To ensure no mistakes were made, substitute [tex]\(x = 2\)[/tex] back into the original equation [tex]\(y = -3(x-1)\)[/tex] and [tex]\(y = x-5\)[/tex]:
1. First equation [tex]\(y = -3(x - 1)\)[/tex]:
[tex]\[
y = -3(2 - 1) = -3(1) = -3
\][/tex]
2. Second equation [tex]\(y = x - 5\)[/tex]:
[tex]\[
y = 2 - 5 = -3
\][/tex]
Since both equations yield [tex]\(y = -3\)[/tex] when [tex]\(x = 2\)[/tex], the solution satisfies both equations. Therefore, there are no additional solutions.
### Conclusion:
The solution to the equation [tex]\(-3(x-1)=x-5\)[/tex] is:
[tex]\[
x = 2
\][/tex]
There are no other solutions. Thus, the final answer is:
[tex]\[
\boxed{2}
\][/tex]
