1963 TMTA High School Algebra I Contest Problem 31

Problem

Solve for $x$ when $\frac{x}{x-1} = 3 + \frac{1}{x-1}.$

$\text{(A)} \quad 2 \quad \text{(B)} \quad -1 \quad \text{(C)} \quad 1 \quad \text{(D)} \quad -2 \quad \text{(E) NOTA}$

Solution

Simplify. $\frac{x}{x-1} = \frac{3(x-1)+1}{x-1}=\frac{3x-2}{x-1}$ Cross multiply. $(x)(x-1)=(3x-2)(x-1)$ $x^2 - x = 3x^2 - 5x + 2$ $0 = 2x^2 - 4x + 2$ This yields $x = 1$ as a solution, but this is extraneous since $1-1=0$ , and the denominator in the fractions given cannot be $0$ , so the answer is $\boxed{\text{(E)}}.$

1963 TMTA High School Mathematics Contests (Problems)
Preceded by Problem 30	TMTA High School Mathematics Contest Past Problems/Solutions	Followed by Problem 32

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1963 TMTA High School Algebra I Contest Problem 31

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