Equation 1: [math]x-3=2-4x[/math] [br]
[table][tr][td]Here is an equation, and then all the steps Clare wrote to solve it: [/td][td][/td][td]Here is the same equation, and the steps Lin wrote to solve it:[/td][/tr][tr][td][math]\displaystyle \begin{align}14x - 2x + 3 &= 3(5x + 9)\\12x + 3& = 3(5x + 9)\\3(4x+1)& = 3(5x + 9)\\4x + 1 &= 5x + 9\\1 &= x + 9\\ \text{-}8 &= x \end{align}[/math][/td][td][/td][td][math]\displaystyle \begin{align}14x - 2x + 3 &= 3(5x + 9)\\12x + 3 &= 3(5x + 9)\\12x + 3 &= 15x + 27\\12x &= 15x + 24\\ \text{-}3x &= 24\\x &= \text{-}8 \end{align}[/math][/td][/tr][/table][br]Are both of their [i]solutions[/i] correct?
Describe some ways the steps they took are alike.
Describe some ways the steps they took are different.
[size=100]Mai and Noah also solved the equation, but some of their steps have errors. Find the incorrect step in each solution and explain why it is incorrect.[br][br][table][tr][td]Mai:[/td][td][/td][td]Noah: [/td][/tr][tr][td][math]\displaystyle \begin{align}14x - 2x + 3 &= 3(5x + 9) \\ 12x + 3 &= 3(5x + 9) \\ 7x + 3 &= 3(9) \\7x + 3 &= 27 \\7x &= 24 \\ x &= \frac{24}{7} \end{align}[/math][/td][td][/td][td][math]\displaystyle \begin{align}14x - 2x + 3 &= 3(5x + 9) \\ 12x + 3 &= 15x + 27 \\ 27x + 3 &= 27 \\ 27x& = 24 \\ x &= \frac{24}{27} \end{align}[/math][/td][/tr][/table][/size]
[math]\frac{12+6x}{3}=\frac{5-9}{2}[/math]
[math]x-4=\frac{1}{3}(6x+54)[/math]
[math]-(3x-12)=9x-4[/math]
I have 24 pencils and 3 cups. The second cup holds one more pencil than the first. The third holds one more than the second. How many pencils does each cup contain?