# Exercise Set 1.1 – No. 1

1. In each part, determine whether the equation is linear in ${{x}}_{{1}}$${{x}}_{{2}}$ and ${{x}}_{{3}}$.

$\large&space;{\color{Blue}&space;(a)\;&space;x_1+5x_2-\sqrt{2}x_3=1}$

Notice that the equation contain $\large&space;{\color{blue}&space;\sqrt2}$, but it is only a constant in front of the variable ${{x}}_{{3}}$.

All the variables have powers 1, and the are no combinations of variables such as ${{x}}_{{1}}{.}{{x}}_{{2}}$, so given equation is linear.

$\large&space;{\color{blue}&space;(b)&space;\;&space;x_1+3x_2+x_1.x_3=2}$

The equation contain term ${{x}}_{{1}}{.}{{x}}_{{3}}$ which is a product of two different variables, so it is not linear.

$\large&space;{\color{blue}&space;(c)&space;\;&space;x_1=-7x_2+3x_3}$

Given equation can be written as:

$\large&space;{\color{blue}&space;\;&space;x_1+7x_2-3x_3=0}$

So, it is linear.

$\large&space;{\color{blue}&space;(d)&space;\;&space;x_1^{-2}+x_2+8x_3=5}$

Given equation contains term $\large&space;{\color{blue}&space;\;&space;x_1^{-2}}$. Since the power of the variable ${{x}}_{{1}}$ is -2. Equation is not linear.

$\large&space;{\color{blue}&space;(e)&space;\;&space;x_1^{{\frac{3}{5}}}-2x_2+x_3=4}$

Given equation contains term $\large&space;{\color{blue}&space;x_1^{{\frac{3}{5}}}}$. Since the power of the variable ${{x}}_{{1}}$ is $\large&space;{\color{blue}&space;{{\frac{3}{5}}}}$. Equation is not linear.

$\large&space;{\color{blue}&space;(f)&space;\;&space;\pi&space;x_1-\sqrt{2}x_2=7^{\frac{1}{3}}}$

Given equation is linear. Same explanation as in part a)