Question

Find the error in the following proof that 2 = 1. Consider the equation a = b. Multiply both sides by a to obtain a² = ab. Subtract b²from both sides to get a² - b² = ab - b². Now factor each side, (a+b) (a-b) = b (a-b),and divide each side by (a-b)to get a + b = bFinally, letequal 1, which shows that 2 = 1

Short answer

By applying all the equations in the form of one then as a result, 2 = 1 is proved.

Step-by-step solution

Step 1:

The equation a = b.

Multiply both sides by a to obtain a² = ab.

Subtract b² from both sides to get a²-b²= ab-b².

Now factor each side, (a+b) (a-b)= b(a-b) and divide each side by(a-b) to get a+b = b Finally, let equal one, which shows that 2 = 1

The equations are defined as,

$$\begin{gathered} a=b \\ a.a=a.b \\ {a}^2=a.b \\ {a}^2-b^2=ab-b^2 \\ (a+b)(a-b)=b(a-b) \\ (a+b=b \\ 1=1 \\ 1=1-1 \\ 1-1=1-1 \\ (1+1\left)\right(1-1)=(1-1\left) \\ \right(1+1)=1 \\ 2=1 \end{gathered}$$

Hence proved.