One-step equations with negatives (Fractions)

Question

I am solving the one-step equation: $$-\frac{3}{4}\alpha = \frac{5}{4}$$ In the answer sheet it says to multiply both sides by the "reciprocal" $-\dfrac{4}{3}$. In one step equations you are supposed to do the opposite on both sides and in this wouldn't the opposite be divide both sides by $-\dfrac{3}{4}$ and not multiply by $-\dfrac{4}{3}$ ? Am I making a mistake or is this just another way of doing it?

​

Answer 1

Multiplying by the "reciprocal" is exactly the same as dividing by the number in front of $\alpha$. In your example it's equivalent to notice that :

$$ \frac{1}{\frac{a}{b}} = \frac{b}{a} $$

Answer 2

If you try it out, you will notice that this will free the $\alpha$ on the LHS. So this operation is the inverse of multiplication with $-3/4$.

Answer 3

You like to get rid of $$ -\frac{3}{4}$$ in$$ -\frac{3}{4} \alpha = \frac{5}{4}$$

Dividing by $$ -\frac{3}{4}$$ means multiplying by $$ -\frac{4}{3}$$ because dividing by $x$ means multiplying by $1/x$

So it is fine to multiply by reciprocal to make our life easier.