Let's solve for [tex]\( t \)[/tex] step-by-step and identify where any algebraic errors might occur.
Given the initial equation:
[tex]\[ w = r_1 t + r_2 t \][/tex]
### Correct Step 1:
Combine like terms on the right side of the equation:
[tex]\[ w = t(r_1 + r_2) \][/tex]
### Correct Step 2:
Isolate [tex]\( t \)[/tex] by dividing both sides of the equation by [tex]\((r_1 + r_2)\)[/tex] (assuming [tex]\( r_1 + r_2 \neq 0 \)[/tex]):
[tex]\[ t = \frac{w}{r_1 + r_2} \][/tex]
### Incorrect Step:
In the given list of steps, the step:
[tex]\[ w = r_1 r_2 t \][/tex]
is algebraically incorrect.
The term [tex]\( r_1 r_2 t \)[/tex] does not correctly represent combining the rates [tex]\( r_1 \)[/tex] and [tex]\( r_2 \)[/tex] when both are working together for the same time duration [tex]\( t \)[/tex]. Instead, [tex]\( w \)[/tex] should be expressed as [tex]\( t(r_1 + r_2) \)[/tex], not [tex]\( r_1 r_2 t \)[/tex].
Therefore, the incorrect step in solving the equation for [tex]\( t \)[/tex] is:
[tex]\[ w = r_1 r_2 t \][/tex]
