Question
Answer
$$(1-x)\cdot(1-y)+2\cdot(1-x)y+3x\cdot(1-y)+4xy=2x+y+1$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}(1-x)\cdot(1-y)+2\cdot(1-x)y+3x\cdot(1-y)+4xy& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}1-y-x+xy+(2-2x)y+3x-3xy+4xy \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}1-y-x+xy+2y-2xy+3x-3xy+4xy \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}-xy-x+y+1+3x-3xy+4xy \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}-4xy+2x+y+1+4xy \xlongequal{ } \\[1 em] & \xlongequal{ } -\cancel{4xy}+2x+y+1+ \cancel{4xy} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}2x+y+1\end{aligned} $$ | |
| ① | Multiply each term of $ \left( \color{blue}{1-x}\right) $ by each term in $ \left( 1-y\right) $. $$ \left( \color{blue}{1-x}\right) \cdot \left( 1-y\right) = 1-y-x+xy $$Multiply $ \color{blue}{2} $ by $ \left( 1-x\right) $ $$ \color{blue}{2} \cdot \left( 1-x\right) = 2-2x $$Multiply $ \color{blue}{3x} $ by $ \left( 1-y\right) $ $$ \color{blue}{3x} \cdot \left( 1-y\right) = 3x-3xy $$ |
| ② | $$ \left( \color{blue}{2-2x}\right) \cdot y = 2y-2xy $$ |
| ③ | Combine like terms: $$ 1 \color{blue}{-y} -x+ \color{red}{xy} + \color{blue}{2y} \color{red}{-2xy} = \color{red}{-xy} -x+ \color{blue}{y} +1 $$ |
| ④ | Combine like terms: $$ \color{blue}{-xy} \color{red}{-x} +y+1+ \color{red}{3x} \color{blue}{-3xy} = \color{blue}{-4xy} + \color{red}{2x} +y+1 $$ |
| ⑤ | Combine like terms: $$ \, \color{blue}{ -\cancel{4xy}} \,+2x+y+1+ \, \color{blue}{ \cancel{4xy}} \, = 2x+y+1 $$ |
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