# How do solve the following linear system?:  7x+2y=2 , 6x+4y=-1 ?

Jan 27, 2016

$\left(\frac{5}{8} , - \frac{19}{16}\right)$

#### Explanation:

There are several approaches to solving this linear system and it is up to you to decide which one you are more comfortable with. In this case, I would show how to solve this by elimination. Know that you also have the option to do the substitution method. Both the elimination and substitution methods are pretty straightforward so I suggest you use any of the two. Another approach would be the graphing method. However it consumes more time particularly because a certain amount of precision in graphing is necessary. Moreover, the process is more complicated especially when the solution to the problem involves fractions (which in this case, it does).

*I will label the equations with 1 and 2 so as to make it easier to explain the process

By Elimination:

$7 x + 2 y = 2$ .................... Equation 1
$6 x + 4 y = - 1$ .............. Equation 2

Multiply the whole Equation 1 by the constant 2. $\left(E q u a t i o n 1\right) \cdot 2$
Then retain Equation 2

$14 x + 4 y = 4$
(-) $6 x + 4 y = - 1$

$8 x = 5$
$x = \frac{5}{8}$

Note that the coefficient of $y$ in both equations are 4. This would allow you to subtract Equation 2 from Equation 1 (or vice versa) to eliminate the $y$ variable from the equation and eventually solve for $x$

Now that we know that $x = \frac{5}{8}$ we can now solve for $y$ using any of the two equations. It will now also be up to you which of the two equations you would use. In this case, you can choose the equation with smaller coefficients or the one that allows you to cancel out some of the bigger coefficients in the equation. For this problem I would show both ways.

*Using Equation 1
$7 \left(\frac{5}{8}\right) + 2 y = 2$
$\frac{35}{8} + 2 y = 2$
$2 y = 2 - \frac{35}{8}$
$2 y = \frac{16}{8} - \frac{35}{8}$
$2 y = - \frac{19}{8}$
$y = - \frac{19}{16}$

*Using Equation 2
$6 \left(\frac{5}{8}\right) + 4 y = - 1$
$\frac{30}{8} + 4 y = - 1$
$\frac{15}{4} + 4 y = - 1$
$4 y = - 1 - \frac{15}{4}$
$4 y = - \frac{19}{4}$
$y = - \frac{19}{16}$

Note that whichever approach you use would lead you to the same final answer. $\left(\frac{5}{8} , - \frac{19}{16}\right)$