# How do you solve using the addition method and determine if the system is independant, dependant, inconsistant given 3/7x+5/9y=27 and 1/9x+2/7y=7?

Feb 6, 2017

$x = 63$ and $y = 0$. The system is consistent and the equations are independent.

#### Explanation:

The system of equations is

$\frac{3}{7} x + \frac{5}{9} y = 27$
$\frac{1}{9} x + \frac{2}{7} y = 7$

Multiply the first equation by $- 7 \text{/} 21$ to eliminate $x$

$- \frac{7}{27} \left(\frac{3}{7} x + \frac{5}{9} y\right) = - \frac{7}{27} \left(27\right)$
$\frac{1}{9} x + \frac{2}{7} y = 7$

This gives

$- \frac{1}{9} x - \frac{35}{243} y = - 7$
$\text{ "1/9x+2/7y" } = 7$

$\frac{241}{1701} y = 0 \implies y = 0$

Plug $y = 0$ back into either of the original equations for $x$

$\frac{3}{7} x + \frac{5}{9} \left(0\right) = 27$
$\frac{3}{7} x = 27$
$x = 27 \left(\frac{7}{3}\right)$
$x = 63$

If the lines intersect, then the system has one solution, given by the point of intersection. The system is consistent and the equations are independent. 