How do you get the following in the equation below?

Question

How do you get the following results in this equation:

$-2a + b - c = 0$

$-3a - 4b + c = 0$

$\frac{a}{1*1-(-4)*(-1)}$ = $\frac{b}{(-3)*(-1)-(-2)*1}$ = $\frac{c}{(-2)*(-4)-(-3)*1}$

$\frac{a}{-3}$ = $\frac{b}{3+2}$ = $\frac{c}{8+3}$

Answer 1

Hint: Attempt to solve the sytem: $$-2a + b - c = 0 \tag1$$

$$-3a - 4b + c = 0 \tag2$$

(1)+(2)

$$-5a-3b=0 \Rightarrow a=\frac{-3b}{5}, b=\frac{-5a}{3} \tag 3$$

$a=\frac{-3b}{5}$ can also be written as $$\frac{a}{-3}=\frac{b}{5} $$

Substitute the value of $a$ in (1):

$$-2(\frac{3b}{5})+b=c$$

$$-2(\frac{-3b}{5})+b=c$$

$$\frac{11}{5}(b)=c \tag4$$

We know $b$ from (3), using that value in (4) $$\frac{11}{5}(\frac{-5a}{3} )=c $$

$$\frac{c}{11}=\frac{a}{-3}$$