Help with simplifying in algebra?

Question

01. Why can't this problem be simplified? --> (5x - 4y) / (5x - y)

02. How do I solve this equation? --> [(4x / 2y) + 2x] / [2x + (2x / y)^2]

Answer 1

Neither of your equations has an = sign so they are impossible to solve.

Answer 2

1.  You can only simplify a fraction if the numerator and denominator have common factors.  Since you can't factor either (5x-4y) or (5x-y), you can't simplify.

Note: a common algebra mistake is to try to add or subtract the same thing from both the numerator and the denominator.  But since 1/2 is not the same as 2/3 (even though you added 1 to both the numerator an the denominator), you can't do that.

2.  I assume you're asking how to simplify the expression.  (It's only an equation if you have an equals sign, unfortunately a lot of math teachers don't emphasize that distinction.)

Follow regular order of operations (carefully!):

First square the (2x/y) and get (4x^2/y^2)

The simplify (4x/2y) by dividing numerator and denominator by 2 and get (2x/y)

You now have [(2x/y)+2x]/[2x + (4x^2/y^2)]

Fractions within fractions can get messy, so let's get rid of them by multiplying each term by y^2:

(2x/y)*y^2 = 2xy

2x*y^2 = 2xy^2

2x*y^2 = 2xy^2 (the same term appears twice)

(4x^2/y^2)*y^2 = 4x^2

We now have: (2xy + 2xy^2) / (2xy^2 + 4x^2)

Factor out 2x from both the numerator and denominator:

[(2x)(y + y^2)] / [(2x)(y^2 + 2x)]

Cancel the (2x)'s:

(y + y^2)/(y^2 + 2x)

Additional Question:

If you can factor the numerator and denominator to a common term, you should because then you can simplify.  Otherwise there's no reason to (unless your teacher says you have to).

Answer 3

1. Because you can only cancel common FACTORS. Neither side can be factored, and the binomials are not the same or opposites, so no cancellation is possible.

2. Reduce and square first:

(2x/y + 2x/1)/(2x/1 + 4x^2/y^2)

Multiply each fraction's top by the LCD, which is y^2: you get:

(2xy + 2xy^2)/(2xy^2 + 4x^2)

Factor out the GCF from each side:

2xy(1+y)/[2x(y^2 + 2x)[

= [y(1+y)]/(y^2 + 2x) after cancelling 2x from each side.

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THird question:

I would factor the top, but not expand the bottom, to see if factor will cancel. In this case there will be no cancellation, because the top is

(2x + 3  )(x +  2 )