Yesterday I started to learn algebra with my own. And the first chapter which I'm learning is Ratio . I'm not getting the red circle marked lines in the below image. I tried to understand it but didn't get it.
Thankyou in advance. and sorry for my bad English. :)
On
Multiply the first equation by $\frac{-a_2}{a_1}$ and add it to the second to get, $$\Big(\frac{-a_2b_1}{a_1} + b_2 \Big)\frac{y}{z} + \frac{-a_2c_1}{a_1} + c_2 = 0$$ $$\Big(\frac{-a_2b_1 + a_1b_2}{a_1}\Big)\frac{y}{z} + \frac{-a_2c_1 + a_1c_2}{a_1}= 0$$ $$\Big(-a_2b_1 + a_1b_2\Big)\frac{y}{z} + -a_2c_1 + a_1c_2= 0$$ $$\Big( a_1b_2 - a_2b_1\Big)\frac{y}{z} = a_2c_1 - a_1c_2$$ $$\frac{y}{z} = \frac{c_1a_2 - c_2a_1}{ a_1b_2 - a_2b_1}$$ $\frac{x}{z}$ can be found in a similar way.
$$\frac{x}{z} = \frac{b_1c_2 - b_2c_1}{a_1b_2 - a_2b_1}$$ \begin{equation} \frac{x}{b_1c_2 - b_2c_1} = \frac{z}{a_1b_2 - a_2b_1} \end{equation} $$\frac{y}{z} = \frac{c_1a_2 - c_2a_1}{a_1b_2 - a_2b_1}$$ \begin{equation} \frac{y}{c_1a_2 - c_2a_1} = \frac{z}{a_1b_2 - a_2b_1} \end{equation} From the above equations we have, \begin{equation} \frac{x}{b_1c_2 - b_2c_1} = \frac{y}{c_1a_2 - c_2a_1} = \frac{z}{a_1b_2 - a_2b_1} \end{equation}
On
The first pair of equations is essentially Cramer's rule applied to the system of equations given above that:
$$ \begin{pmatrix}a_1&b_1\\a_2&b_2\end{pmatrix}\cdot\begin{pmatrix}x/z\\y/z\end{pmatrix}=\begin{pmatrix}-c_1\\-c_2\end{pmatrix}\implies \frac{x}{z}=\frac{\begin{vmatrix}-c_1&b_1\\-c_2&b_2\end{vmatrix}}{\begin{vmatrix}a_1&b_1\\a_2&b_2\end{vmatrix}},\;\frac{y}{z}=\frac{\begin{vmatrix}a_1&-c_1\\a_2&-c_2\end{vmatrix}}{\begin{vmatrix}a_1&b_1\\a_2&b_2\end{vmatrix}},$$
To obtain the second circled equation, observe that the first circled pair has $z$ in the denominator of the left hand side, and $a_1b_2-a_2b_1$ in the denominator on the right. Multiplying both equations by $z$ and dividing by the numerator on the right you get $\frac{z}{a_1b_2-a_2b_1}$ as the right hand side of both equations, so you can combine them into a single equation.
The first set of equations comes from solving the two equations with two unknowns right above. The book presumably has a section on how to do that.
The second set comes from the first by multiplying with $z$ and diving by $b_1c_2-b_2c_1$.