# Faster way to work out multiples

#### uperkurk

Hi all, my fractions are not great to be honest but here is my question.

Using addition, solve for a and b:

$$\displaystyle 4a-6b=15$$
$$\displaystyle 6a-4b=10$$

So I find a common multiple of b, I choose to use 24.

$$\displaystyle 16a-24b=60$$
$$\displaystyle 36a-24b=60$$

$$\displaystyle 16a-24b=60$$
$$\displaystyle -36a+24b=-60$$

I'm left with: $$\displaystyle -20a=0 == a=0$$

Is this correct so far? If so then I plug the answer back into the equation and get:

$$\displaystyle 4-6b=15$$

I only know from using a website that $$\displaystyle b=-\frac{5}{2}$$

My question is I just don't know how to go about finding the correct fraction, is there a simple method?

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#### Kanwar245

After you've found your answers plug them back in the two equations and see if LHS = RHS

#### uperkurk

I don't know what LHS = RHS means. Could you elaborate?

#### Kanwar245

LHS means left hand side, RHS means right hand side

#### uperkurk

But my question was with regards to finding the quick way to the correct fraction

#### Paze

Hi Uperkurk.

Your answer also seems to be correct. You are left with $$\displaystyle a=0$$ So by plugging that into either one of your equations (I'm choosing the first one for my example), you get $$\displaystyle 4\cdot 0-6b\right)=15$$ It seems that by solving this equation you get $$\displaystyle b=\frac{-5}{2}$$ which is harmonious with the answer.

For a more in-depth view on how to solve the equation

$$\displaystyle \\4\cdot 0-6b=15\\\\0-6b=15\\\\-6b=15\\\\b=\frac{15}{-6}\\\\b=-\frac{5}{2}$$

Does this clear it up for you?

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#### uperkurk

Hi Uperkurk.

Your answer also seems to be correct. You are left with $$\displaystyle a=0$$ So by plugging that into either one of your equations (I'm choosing the first one for my example), you get $$\displaystyle 4\cdot 0-6b\right)=15$$ It seems that by solving this equation you get $$\displaystyle b=\frac{-5}{2}$$ which is harmonious with the answer.
$$\displaystyle \\4\cdot 0-6b=15\\\\0-6b=15\\\\-6b=15\\\\b=\frac{15}{-6}\\\\b=-\frac{5}{2}$$