Answer
The solution set is
$\left\{ \left (\frac{b_2c_1-b_1c_2}{a_1b_2-a_2b_1},\frac{a_1c_2-a_2c_1}{a_1b_2-a_2b_1} \right ) \right\}$.
Work Step by Step
The given system of equations are
$\Rightarrow a_1x+b_1y=c_1$ ...... (1)
$\Rightarrow a_2x+b_2y=c_2$ ...... (2)
Multiply equation (1) by $-a_2$.
$\Rightarrow -a_2(a_1x+b_1y)=-a_2(c_1)$
Apply distributive property.
$\Rightarrow -a_2a_1x-a_2b_1y=-a_2c_1$ ...... (3)
Multiply equation (2) by $a_1$.
$\Rightarrow a_1(a_2x+b_2y)=a_1(c_2)$
Apply distributive property.
$\Rightarrow a_1a_2x+a_1b_2y=a_1c_2$ ...... (4)
Add equation (3) and (4).
$\Rightarrow -a_2a_1x-a_2b_1y+a_1a_2x+a_1b_2y=-a_2c_1+a_1c_2$
Add like terms.
$\Rightarrow -a_2b_1y+a_1b_2y=-a_2c_1+a_1c_2$
Apply distributive property.
$\Rightarrow (-a_2b_1+a_1b_2)y=-a_2c_1+a_1c_2$
Isolate $y$.
$\Rightarrow y=\frac{-a_2c_1+a_1c_2}{-a_2b_1+a_1b_2}$
Or we can write.
$\Rightarrow y=\frac{a_1c_2-a_2c_1}{a_1b_2-a_2b_1}$
Substitute the value of $y$ into equation (1).
$\Rightarrow a_1x+b_1\left ( \frac{a_1c_2-a_2c_1}{a_1b_2-a_2b_1}\right )=c_1$
Apply distributive property.
$\Rightarrow a_1x+\frac{b_1a_1c_2-b_1a_2c_1}{a_1b_2-a_2b_1}=c_1$
Subtract $\frac{b_1a_1c_2-b_1a_2c_1}{a_1b_2-a_2b_1}$ from both sides.
$\Rightarrow a_1x+\frac{b_1a_1c_2-b_1a_2c_1}{a_1b_2-a_2b_1}-\frac{b_1a_1c_2-b_1a_2c_1}{a_1b_2-a_2b_1}=c_1-\frac{b_1a_1c_2-b_1a_2c_1}{a_1b_2-a_2b_1}$
Simplify.
$\Rightarrow a_1x=c_1-\frac{b_1a_1c_2-b_1a_2c_1}{a_1b_2-a_2b_1}$
Multiply and divide $c_1$ by $(a_1b_2-a_2b_1)$.
$\Rightarrow a_1x=\frac{(a_1b_2-a_2b_1)}{(a_1b_2-a_2b_1)}\cdot c_1-\frac{b_1a_1c_2-b_1a_2c_1}{a_1b_2-a_2b_1}$
Simplify.
$\Rightarrow a_1x=\frac{c_1(a_1b_2-a_2b_1)}{(a_1b_2-a_2b_1)}-\frac{b_1a_1c_2-b_1a_2c_1}{a_1b_2-a_2b_1}$
Apply distributive property.
$\Rightarrow a_1x=\frac{c_1a_1b_2-c_1a_2b_1}{a_1b_2-a_2b_1}-\frac{b_1a_1c_2-b_1a_2c_1}{a_1b_2-a_2b_1}$
Add numerator because denominators are equal.
$\Rightarrow a_1x=\frac{c_1a_1b_2-c_1a_2b_1-b_1a_1c_2+b_1a_2c_1}{a_1b_2-a_2b_1}$
Add like terms.
$\Rightarrow a_1x=\frac{c_1a_1b_2-b_1a_1c_2}{a_1b_2-a_2b_1}$
Divide both sides by $a_1$.
$\Rightarrow \frac{a_1x}{a_1}=\frac{c_1a_1b_2-b_1a_1c_2}{a_1(a_1b_2-a_2b_1)}$
Factor out $a_1$ from the numerator.
$\Rightarrow \frac{a_1x}{a_1}=\frac{a_1(c_1b_2-b_1c_2)}{a_1(a_1b_2-a_2b_1)}$
Cancel common terms.
$\Rightarrow x=\frac{c_1b_2-b_1c_2}{a_1b_2-a_2b_1}$
Rearrange.
$\Rightarrow x=\frac{b_2c_1-b_1c_2}{a_1b_2-a_2b_1}$.
