Answer
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{n\frac{(k-2)n-(k-4)}{2}-n\frac{(m-2)n-(m-4)}{2}}{k-m}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{(\frac{(k-2)n-(k-4)}{2}-\frac{(m-2)n-(m-4)}{2})n}{k-m} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{((k-2)n-(k-4)-((m-2)n-(m-4)))\cdot\frac{1}{2}n}{k-m} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{(1kn-2n-(k-4)-(1mn-2n-(m-4)))\cdot\frac{1}{2}n}{k-m} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} \htmlClass{explanationCircle explanationCircle6}{\textcircled {6}} } }}}\frac{(1kn-2n-k+4-(1mn-2n-m+4))\cdot\frac{1}{2}n}{k-m} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle7}{\textcircled {7}} } }}}\frac{(1kn-2n-k+4-mn+2n+m-4)\cdot\frac{1}{2}n}{k-m} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle8}{\textcircled {8}} } }}}\frac{(1kn-mn-k+m)\cdot\frac{1}{2}n}{k-m} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle9}{\textcircled {9}} } }}}\frac{\frac{kn-mn-k+m}{2}n}{k-m} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle10}{\textcircled {10}} } }}}\frac{\frac{kn^2-mn^2-kn+mn}{2}}{k-m} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle11}{\textcircled {11}} } }}}\frac{kn^2-mn^2-kn+mn}{2k-2m}\end{aligned} $$ | |
| ① | Use the distributive property. |
| ② | Use the distributive property. |
| ③ | $$ \left( \color{blue}{k-2}\right) \cdot n = kn-2n $$ |
| ④ | $$ \left( \color{blue}{m-2}\right) \cdot n = mn-2n $$ |
| ⑤ | Remove the parentheses by changing the sign of each term within them. $$ - \left( k-4 \right) = -k+4 $$ |
| ⑥ | Remove the parentheses by changing the sign of each term within them. $$ - \left( m-4 \right) = -m+4 $$ |
| ⑦ | Remove the parentheses by changing the sign of each term within them. $$ - \left( mn-2n-m+4 \right) = -mn+2n+m-4 $$ |
| ⑧ | Combine like terms: $$ kn \, \color{blue}{ -\cancel{2n}} \,-k+ \, \color{green}{ \cancel{4}} \,-mn+ \, \color{blue}{ \cancel{2n}} \,+m \, \color{green}{ -\cancel{4}} \, = kn-mn-k+m $$ |
| ⑨ | Multiply $kn-mn-k+m$ by $ \dfrac{1}{2} $ to get $ \dfrac{ kn-mn-k+m }{ 2 } $. Step 1: Write $ kn-mn-k+m $ as a fraction by putting $ \color{red}{1} $ in the denominator. Step 2: Multiply numerators and denominators. Step 3: Simplify numerator and denominator. $$ \begin{aligned} kn-mn-k+m \cdot \frac{1}{2} & \xlongequal{\text{Step 1}} \frac{kn-mn-k+m}{\color{red}{1}} \cdot \frac{1}{2} \xlongequal{\text{Step 2}} \frac{ \left( kn-mn-k+m \right) \cdot 1 }{ 1 \cdot 2 } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ kn-mn-k+m }{ 2 } \end{aligned} $$ |
| ⑩ | Multiply $ \dfrac{kn-mn-k+m}{2} $ by $ n $ to get $ \dfrac{ kn^2-mn^2-kn+mn }{ 2 } $. Step 1: Write $ n $ as a fraction by putting $ \color{red}{ 1 } $ in the denominator. Step 2: Multiply numerators and denominators. Step 3: Simplify numerator and denominator. $$ \begin{aligned} \frac{kn-mn-k+m}{2} \cdot n & \xlongequal{\text{Step 1}} \frac{kn-mn-k+m}{2} \cdot \frac{n}{\color{red}{1}} \xlongequal{\text{Step 2}} \frac{ \left( kn-mn-k+m \right) \cdot n }{ 2 \cdot 1 } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ kn^2-mn^2-kn+mn }{ 2 } \end{aligned} $$ |
| ⑪ | Divide $ \dfrac{kn^2-mn^2-kn+mn}{2} $ by $ k-m $ to get $ \dfrac{ kn^2-mn^2-kn+mn }{ 2k-2m } $. Step 1: To divide rational expressions, multiply the first fraction by the reciprocal of the second fraction. Step 2: Multiply numerators and denominators. Step 3: Simplify numerator and denominator. $$ \begin{aligned} \frac{ \frac{kn^2-mn^2-kn+mn}{2} }{k-m} & \xlongequal{\text{Step 1}} \frac{kn^2-mn^2-kn+mn}{2} \cdot \frac{\color{blue}{1}}{\color{blue}{k-m}} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \left( kn^2-mn^2-kn+mn \right) \cdot 1 }{ 2 \cdot \left( k-m \right) } \xlongequal{\text{Step 3}} \frac{ kn^2-mn^2-kn+mn }{ 2k-2m } \end{aligned} $$ |