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Problem 700

AMC 10/12, early questions

Algebra Difficulty 3.9 Multiple choice National Math Olympiad in Slovenia · Slovenia

Let $a$ and $b$ be two different real numbers. For which $x$ does the equality $\frac{x-a}{x-b} = \frac{x-b}{x-a}$ hold?

Pick one

Official solution

Multiplying the given equality by $(x-a)(x-b)$ , we get $(x-a)^2 = (x-b)^2$ . Squaring both sides and subtracting $x^2$ we get $-2a x + a^2 = -2b x + b^2$ , which can be further rearranged into $2(b-a)x = (b-a)(b+a)$ . Since $b-a \neq 0$ , we can divide both sides of the equality by $b-a$ to get $2x = b+a$ . From here we get $x = \frac{a+b}{2}$ .

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