Answer
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{\frac{1}{s-5}}{1+\frac{1}{s-5}(s-a)}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{\frac{1}{s-5}}{1+\frac{-a+s}{s-5}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{\frac{1}{s-5}}{\frac{-a+2s-5}{s-5}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{1}{-a+2s-5}\end{aligned} $$ | |
| ① | Multiply $ \dfrac{1}{s-5} $ by $ s-a $ to get $ \dfrac{-a+s}{s-5} $. Step 1: Write $ s-a $ as a fraction by putting $ \color{red}{ 1 } $ in the denominator. Step 2: Multiply numerators and denominators. Step 3: Simplify numerator and denominator. $$ \begin{aligned} \frac{1}{s-5} \cdot s-a & \xlongequal{\text{Step 1}} \frac{1}{s-5} \cdot \frac{s-a}{\color{red}{1}} \xlongequal{\text{Step 2}} \frac{ 1 \cdot \left( s-a \right) }{ \left( s-5 \right) \cdot 1 } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ s-a }{ s-5 } = \frac{-a+s}{s-5} \end{aligned} $$ |
| ② | Add $1$ and $ \dfrac{-a+s}{s-5} $ to get $ \dfrac{ \color{purple}{ -a+2s-5 } }{ s-5 }$. Step 1: Write $ 1 $ as a fraction by putting $ \color{red}{ 1 } $ in the denominator. Step 2: To add raitonal expressions, both fractions must have the same denominator. |
| ③ | Divide $ \dfrac{1}{s-5} $ by $ \dfrac{-a+2s-5}{s-5} $ to get $ \dfrac{ 1 }{ -a+2s-5 } $. Step 1: To divide rational expressions, multiply the first fraction by the reciprocal of the second fraction. Step 2: Cancel $ \color{red}{ s-5 } $ in first and second fraction. Step 3: Multiply numerators and denominators. Step 4: Simplify numerator and denominator. $$ \begin{aligned} \frac{ \frac{1}{s-5} }{ \frac{\color{blue}{-a+2s-5}}{\color{blue}{s-5}} } & \xlongequal{\text{Step 1}} \frac{1}{s-5} \cdot \frac{\color{blue}{s-5}}{\color{blue}{-a+2s-5}} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{1}{\color{red}{1}} \cdot \frac{\color{red}{1}}{-a+2s-5} \xlongequal{\text{Step 3}} \frac{ 1 \cdot 1 }{ 1 \cdot \left( -a+2s-5 \right) } = \\[1ex] & \xlongequal{\text{Step 4}} \frac{ 1 }{ -a+2s-5 } \end{aligned} $$ |