How do solve the following linear system?:  -x-7y=14 , 3x-2y=2 ?

Jul 6, 2017

Isolate one of the variables in the one of the equations and use that to solve the other variable. Once one of the variable is defined, use it to define the other variable.

Explanation:

Solving linear equations implies finding the intersections of the two equations.

There are many ways to solve a linear system given two equations. In my opinion, the easiest is to isolate one of the variables in one equation, and sub it into the other.

We'll isolate the first equation since it's almost done.

$- x - 7 y = 14$

$- x = 14 + 7 y$

$x = - 14 - 7 y$

Now we sub in $x = - 14 - 7 y$ into the other equation.

$3 x - 2 y = 2$

$3 \left(- 14 - 7 y\right) - 2 y = 2$

Now we expand the bracket.

$\left(- 42 - 21 y\right) - 2 y = 2$

$- 42 - 21 y - 2 y = 2$

And we add like terms.

$- 21 y - 2 y = 2 + 42$

$- 23 y = 44$

Isolate $y$.

$y = - \frac{44}{23}$

Now we use this value to solve for $x$. We can use any equation, as long as it's the original two.

$- x - 7 y = 14$

$- x - 7 \left(- \frac{44}{23}\right) = 14$

$- x + \frac{308}{23} = 14$

$- x = 14 - \frac{308}{23}$

$x = \frac{308}{23} - 14$

$x = - \frac{14}{23}$

Thus, the linear system intersects at $\left(- \frac{14}{23} , - \frac{44}{23}\right)$.

We can double check our work by graphing both equations and finding their intersection.

graph{(-x-7y-14)(3x-2y-2)=0}

Hope this helps :)