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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. :)

Image

Yogesh Tripathi
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3 Answers3

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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}

user222031
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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.

MvG
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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 $b_1c_2-b_2c_1$.

Henrik supports the community
  • 6,831
  • it's not given. Can you explain me please – Yogesh Tripathi Mar 30 '15 at 22:17
  • You're probably going to run into other things that the author of the book assumes you know about, so I would advise you to find a book that's more suited for your level. – Henrik supports the community Mar 30 '15 at 22:19
  • can you tell me from which book I can learn such type of problems.Your advice will be very helpful. – Yogesh Tripathi Mar 30 '15 at 22:22
  • I basically only know the books I used at university, and that's 12-20 years ago, so many of then probably can't be bought anymore, but for that stuff it was "Linear Algebra" by Robert Messer. My advice would be to ask someone closer, they might also have an understanding of your level, and might be able to recommend books in your native language. – Henrik supports the community Mar 30 '15 at 22:29