Beth is solving this equation: 1/x + 3 = 3/x

She says “I can multiply both sides by x and get the linear equation 1 + 3x = 3, whose solution is x = 2/3.”

Which of the following statements makes this a correct argument, or shows that it is incorrect? Select all that apply.

A). The equation is not linear, so you cannot use the methods normally used for solving linear equations.

B). You can assume x ≠ 0 because both sides are undefined if x = 0.

C). After multiplying both sides by x you need to subtract 1 from both sides.

D). You cannot multiply both sides by x because you do not know what x is.

The really tough one to get rid of is A. She's right in her method and it sounds like A should be a deal breaker. Suppose however, you make y = 1/x. Now the equation is linear.

y + 3 = 3y Subtract y from both sides

3 = 3y - y

3 = 2y Now divide this by by 2

3/2 = y

But they are not the same answer!! You've gotten rid of one problem only to create another. Well yes. I never said it was easy getting rid of A. But remember y = 1/x, so x will equal 1 / y.

x = 1/y = 1//3/2 = 2/3

Now they agree.

That's kind of sneaky you might say. That's right it is. But its not wrong. A is not part of the answer.

B is certainly right.

C is correct as well, That's the way the equation is ultimately solved.

D is nonsense. You can do all sorts of things with the letters in equations