Factoring?

[flat dragon]

My Teacher sent me this, but I don't understand it well. Even when my teacher explained it. Can someone help me solve this?

[flat dragon]

I don't want to use chat Gpt because I still don't understand

[ashen barn]

You need to use the equality in order to find the expression

[balmy kayak]

Fair enough. My first step would be to expand both the left hand and right hand side of the equality.

That becomes x + /sqrt (xy) = 3 /sqrt (xy) + 15y.

This can be simplified to x = 2/sqrt(xy) + 15y. I’ll call this Equation 1.

What you would like to do is try and simplify by eliminating some things. So now I would replace x with the right hand side of Equation 1.

I think from there you might be able to solve it. Does this make sense @flat dragon

[flat dragon]

Oh ok thanks

[river sentinel]

Did your teacher give you the answer?
If yes, what answer did your teacher give you?

[river sentinel]

Nice steps.
But, is it not better to substitute for the radical, $\sqrt{xy}$ rather than substituting for the variable, $x$? What do you think?

[surreal ice]

The Mathematics Guy

[balmy kayak]

Perhaps you’re right. Both methods should work fine but substituting the radical might be a more elegant approach

I just chose the x term for simplicity in terms of the coefficients

[ornate stump]

expand both sides, we get: $x + \sqrt{xy} = 3\sqrt{xy} + 15y$

[surreal ice]

1 divided by 0 equals Infinity

[ornate stump]

so $x + 15y = 2\sqrt{xy}$

[surreal ice]

1 divided by 0 equals Infinity

[ornate stump]

or $\sqrt{xy} = \frac{x + 15y}{2}$

[surreal ice]

1 divided by 0 equals Infinity

[ornate stump]

sub that in the equation

$\frac{2x + \sqrt{xy} + 3y}{x + \sqrt{xy} - y}$

[surreal ice]

1 divided by 0 equals Infinity

[ornate stump]

so $2x + \sqrt{xy} + 3y = 2x + \frac{x}{2} + \frac{15y}{2} + 3y$

[surreal ice]

1 divided by 0 equals Infinity

[ornate stump]

$= \frac{5x}{2} + \frac{21y}{2}$

[surreal ice]

1 divided by 0 equals Infinity

[ornate stump]

that's the numerator

denominator $x + \sqrt{xy} - y = x + \frac{x + 15y}{2} - y$

[surreal ice]

1 divided by 0 equals Infinity

[ornate stump]

$= \frac{3x}{2} + \frac{13y}{2}$

[surreal ice]

1 divided by 0 equals Infinity

[ornate stump]

so since we need to find the ratio between the numerator and denominator

then it is not different as finding the ratio between 2numerator and 2 denominator

or $(5x + 21y) : (3x + 13y)$ if i calculated all correctly

[surreal ice]

1 divided by 0 equals Infinity

[flat dragon]

Not yet lol, he just said to try to solve the problem. for "fun"

[river sentinel]

Are you good with the question?
Would you like to see my explanation/solution?

[ornate stump]

ignore my previous attempt

i was wrong

i got $\frac{5x - 9y}{3x - 13y}$

[surreal ice]

1 divided by 0 equals Infinity

[ornate stump]

probably he's trolling you

[river sentinel]

I'm wondering if I made any error.
The numerator is the same, but I got a different denominator from yours.
Number (7.) on https://successexams.appspot.com/MATHEMATICS/ExpressionsEquations.html#one-twenty

[flat dragon]

Yes

[flat dragon]

Oh that's actually made sense

Yo tysm😭

[radiant dew]

is that it or are we supposed tto find a numerical val.?

@flat dragon

[flat dragon]

Nah it's ok

Thank you guys