Question

Which of the following pairs are equivalent expressions?

A. `3x +1/4-x+1 1/2` and `4x+1 3/4`
B. `2=6x` and `2(3x+1)`
C. `3(x-1)-(1+x)` and `2x+3`
D. `3x` and `4(x+1) - x - 4`
E. `5.5+2.1x+3.8x-4.1` and `1.4+5.9x`

Answer 1

The options which have equal expressions are B, D, and E.

Explanation:

Here are all the options written out:

A. `3x +1/4-x+1 1/2` and `4x+1 3/4`
B. `2+6x` and `2(3x+1)`
C. `3(x-1)-(1+x)` and `2x+3`
D. `3x` and `4(x+1) - x - 4`
E. `5.5+2.1x+3.8x-4.1` and `1.4+5.9x`

Note: the first expression in option B was written as `2=6x,` so `2+6x` is the assumed intention.

To check if each of these options has a pair of equal expressions, we can do one of two things:

1. Set the two expressions "equal" and see if we can reduce the equality to a tautology (something obviously true, like `3=3`)
2. Try to simplify one of the expressions until it's the same as the other.

Typically, the easiest thing to do is start with the more complicated expression, and see if it can be simplified into the shorter one. For example, using option A:

`color(white)= 3x +1/4-x+1 1/2`

`=3x-x+1/4+1 1/2` (by rearranging terms)

`="    "2x"   " + 1/4 + 1 1/2` (by combining the `x` terms)

`="    "2x"   " + 1/4 + 1 2/4` (using equivalent fractions)

`="    "2x"   " + "    "1 3/4"     "` (by combining the constants)

Now that we've simplified the left expression as far as we can, we ask: is this equal to the right expression? In other words, is it true that `2x+1 3/4 = 4x+ 1 3/4?`

The answer should be clear: it's not. There are 2 `x"'s"` on the left side, but 4 `x"'s"` on the right. So, the two expressions are not equal. We would not select option A.

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Once you get the hang of doing it this way, there's a shortcut you can use. When two expressions are equal, their corresponding terms are also equal. In other words, they have the same number of `x"'s"`, and the same constants. As an example:

`ax+b = cx+d` if and only if `ax=cx` and `b=d.`

Using option C this time, we check: how many `x"'s"` are in the first expression `color(blue)3(color(blue)x-1)color(orange)-(1+color(orange)x)?` It has `color(blue)(3x) color(orange)(- x)`, which is `2x`. This matches the `2x` in the 2nd expression `2x+3.`

Now, what is the constant term in the first expression? It has a constant of `3(–1)-1,` which is `-3-1`, or `–4`. This does not match the constant of `3` in the 2nd expression. So, these two expressions are not equal. We would not select option C.