# Math Help - algebra

1. ## algebra

I'm helping my little sister with her HW and its been years since i've seen this stuff...

How would I solve
z-6 - 5z
2z+3 2z+3

2. The two fractions have the same denominator.

3. Originally Posted by Rakaa
I'm helping my little sister with her HW and its been years since i've seen this stuff...

How would I solve
z-6 - 5z
2z+3 2z+3
You are wanting to simplify? (solve implies that you want to find the value of Z, but you can't do that if you don't know what it is equal to)

4. Originally Posted by angel.white
You are wanting to simplify? (solve implies that you want to find the value of Z, but you can't do that if you don't know what it is equal to)
Yes I want to simplify

5. Originally Posted by Rakaa
I'm helping my little sister with her HW and its been years since i've seen this stuff...

How would I solve
z-6 - 5z
2z+3 2z+3
$\frac {z-6}{2z+3} - \frac {5z}{2z+3}$

They both have the same denominator, so you can combine them:

$=\frac {z-6-5z}{2z+3}$

z and -5z have the same variable so you could look at it like this: z -z -z -z -z -z
it should be clear then that they combine to equal -4z

$=\frac {-4z-6}{2z+3}$

both terms in the numerator are multiples of -2, so factor out a -2

$=\frac {-2(2z+3)}{2z+3}$

And since -2 is multiplied and divided by 2z+3, the 2z+3 cancels out to equal one

$=-2*1$

Anything times 1 is itself

$=-2$

6. How would I do

y-3 - 2y-7
y+5 y+5

It would become
y-3 - 2y-7
y+5

Then

-1y-(-4)
y+5

And then I become totally lost...

7. $\frac{y-3}{y+5} - \frac{2y - 7}{y+5}$
$= \frac{y - 3 {\color{red} \: - \:} (2y - 7)}{y + 5}$
$= \frac{y - 3 {\color{red} \: - \:}2y {\color{red} \: - \:} (-7)}{y+5}$

Negative of a negative = Positive

8. Originally Posted by Rakaa
How would I do

y-3 - 2y-7
y+5 y+5

It would become
y-3 - 2y-7
y+5

Then

-1y-(-4)
y+5

And then I become totally lost...
The minus sign applies to both terms look at it like this:

$\frac {y-3}{y+5} - \frac {2y-7}{y+5}$

combine

$\frac {y-3-(2y-7)}{y+5}$

distribute the minus sign

$\frac {y-3-2y+7}{y+5}$

combine like terms

$\frac {-y+4}{y+5}$

There are other ways to simplify from here, but "simplify" is kind of ambiguous, this answer is probably what her book is looking for.

Anyway, with that negative sign, it applies to the whole term. You could think of it like this also:
$\frac {y-3}{y+5} +(-1)* \frac {2y-7}{y+5}$

Or it may help to work backwards:
$\frac {-2y+7}{y+5}$

= $\frac {(-1)2y+(-1)(-1)7}{y+5}$

= $\frac {(-1)(2y+(-1)7)}{y+5}$

= $\frac {(-1)(2y-7)}{y+5}$

= $(-1)\frac {(2y-7)}{y+5}$

= $-\frac {2y-7}{y+5}$

However you want to look at it, just understand the sign out front needs to get distributed.

9. Thanks a million..

I really wish I wouldve kept my mind sharp with all of this...

z+6 + 3z2+19z+19
z+5 z+5

next

z+6 + 3z2+19z+19
z+5

29z+26
z+5

Whats the next step? I'm still semi confused about that..

10. Originally Posted by Rakaa
Thanks a million..

I really wish I wouldve kept my mind sharp with all of this...

z+6 + 3z2+19z+19
z+5 z+5

next

z+6 + 3z2+19z+19
z+5

29z+26
z+5

Whats the next step? I'm still semi confused about that..
with this one, you cannot combine $z^2$ and z. Another way of looking at it which might help is to factor out the Z.

so z+19z = z(1+19) = 20z

but $3z^2 + 19z = z(3z+19)$
3z and 19 are not like terms. So $z^2$ and z are not like terms.

You would do it like this:

$\frac {z+6+3z^2+19z+19}{z+5}$

$= \frac {3z^2+20z+25}{z+5}$

Split up the numerator

$= \frac {3z^2+15z+5z+25}{z+5}$

group them (you don't have to do all these steps I'm doing, but I want to hit every step along the way to make it easier to see)

$= \frac {(3z^2+15z)+(5z+25)}{z+5}$

Factor out a 3z

$= \frac {3z(z+5)+(5z+25)}{z+5}$

Factor out a 5

$= \frac {3z(z+5)+5(z+5)}{z+5}$

Factor out a (z+5)

$= \frac {(z+5)(3z+5)}{z+5}$

z+5 is in the numerator and the denominator, so it cancels out

$= 3z+5$

11. One more.. I'm starting to get it..

4y2+zy-3 - 3y2-2y-4
5y+1 z+5

12. Originally Posted by Rakaa
One more.. I'm starting to get it..

4y2+zy-3 - 3y2-2y-4
5y+1 z+5
The only difference here is that you don't have a common denominator, so you will have to make one before you can combine the fractions.

In simple arithmetic, you combine fractions without a common denominator by first finding the least common multiple of both denominators, and then multiplying the numerator and denominator of both fractions by the necessary amount to change the denominator to that multiple. For example:

$\frac12 + \frac23$

The least common multiple of 2 and 3 is 6, so we do

$\frac12 + \frac23$

$=\frac{1\color{red}\cdot3}{2\color{red}\cdot3} + \frac{2\color{red}\cdot2}{3\color{red}\cdot2}$

$=\frac36 + \frac46 = \frac{3 + 4}6 = \frac76$

This basic process doesn't change with the introduction of variable expressions. We have:

$\frac{4y^2+zy-3}{5y+1} - \frac{3y^2-2y-4}{z+5}$

The least common multiple of $5y + 1$ and $z + 5$ is $(5y + 1)(z + 5)$, so we simplify thus:

$\frac{4y^2+zy-3}{5y+1} - \frac{3y^2-2y-4}{z+5}$

$=\frac{\left(4y^2+zy-3\right){\color{red}(z + 5)}}{(5y+1){\color{red}(z + 5)}} - \frac{\left(3y^2-2y-4\right){\color{red}(5y + 1)}}{{\color{red}(5y + 1)}(z+5)}$

$=\frac{\left(4y^2+zy-3\right)(z + 5) - \left(3y^2-2y-4\right)(5y + 1)}{(5y + 1)(z+5)}$

You should be able to take it from here.