Algebra I

Let a=b a = b .

Then, 1ab=a22abb2=a2b23b(ab)=(a+b)(ab)4b=a+b5b=b+b6b=2b71=2. \begin{array} {c r c l } \color{#D61F06} {1} & ab & = & a^2 \\ \color{#EC7300} {2} & ab - b^2 & = & a^2 -b^2 \\ \color{gold} {3} & b(a-b) & = & (a+b)(a-b) \\ \color{#20A900} {4} & b & = & a+b \\ \color{#3D99F6} {5} & b & = & b+b \\ \color{#302B94} {6} & b & = & 2b \\ \color{#69047E} {7} & 1 & = & 2. \end{array} Which line is the first that contains an error in this proof?

When 1 = 2

         

Let's consider the transition from line 1 to 2: 1ab=a22abb2=a2b2. \begin{array} {c r c l } \color{#D61F06} {1} & ab & = & a^2 \\ \color{#EC7300} {2} & ab - b^2 & = & a^2 -b^2. \end{array} b2b^2 is subtracted from both sides of the equal sign. Is it equally valid to do the operation below instead? 1ab=a22b2ab=b2a2 \begin{array} {c r c l } \color{#D61F06} {1} & ab & = & a^2 \\ \color{#EC7300} {2} & b^2 - ab & = & b^2 - a^2 \\ \end{array}

When 1 = 2

         

2abb2=a2b23b(ab)=(a+b)(ab) \begin{array} {c r c l } \color{#EC7300} {2} & ab - b^2 & = & a^2 -b^2 \\ \color{gold} {3} & b(a-b) & = & (a+b)(a-b) \\ \end{array}

On the left-hand side of line 3, the value bb is "factored out" from abb2 ab-b^2 to get b(ab). b(a-b) .

Is the equation below true? abb2a2=ab(1ba) ab - b^2 - a^2 = ab(1-b-a)

When 1 = 2

         

2abb2=a2b23b(ab)=(a+b)(ab) \begin{array} {c r c l } \color{#EC7300} {2} & ab - b^2 & = & a^2 -b^2 \\ \color{gold} {3} & b(a-b) & = & (a+b)(a-b) \\ \end{array}

Changing a2b2 a^2 - b^2 to (a+b)(ab) (a+b)(a-b) is known as the difference of squares identity (it was first introduced in a previous course, as shown in the video below). What if we started with fourth powers instead, that is, a4b4? a^4 - b^4 ? Which expression is equivalent?

When 1 = 2

         

Let's go back to the moment of mistake: 3b(ab)=(a+b)(ab)4b=a+b. \begin{array} {c r c l } \color{gold} {3} & b(a-b) & = & (a+b)(a-b) \\ \color{#20A900} {4} & b & = & a+b. \end{array} (ab) (a-b) does what is known as a "cancel" on both sides of the equal sign. Since the problem set up a=b, a = b, this implies division by zero. Suppose that ab a \neq b instead; what identity would be used to cancel? ((Assume Q0.)Q \neq 0.)

When 1 = 2

         
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