# How do you solve the system of equations by graphing and then classify the system as consistent or inconsistent 5x-2y=20 and -2x+y=-10?

Oct 7, 2016

Consistent system of equations

Point of intersection is:$\text{ } \left(x , y\right) = \left(0 , - 10\right)$

#### Explanation:

If we have an inconsistent system of equation then there is no solution. So lets have a play and see what we have got:

Given:
$5 x - 2 y = 20 \text{ ............................} E q u a t i o n \left(1\right)$
$- 2 x + y = - 10 \text{ ......................} E q u a t i o n \left(2\right)$

Rearranging these gives:

$y = \frac{5}{2} x - 10 \text{ .............................} E q u a t i o n \left({1}_{a}\right)$
$y = \textcolor{w h i t e}{.} 2 x - 10 \text{ ..............................} E q u a t i o n \left({2}_{a}\right)$

Note that both equations have the constant of -10. This constant is where they cross the y-axis. As they are both have the same value then it is a shared point. The lines cross the y-axis at $x = 0$ so the solution has to be $\left(x , y\right) = \left(0 , - 10\right)$

'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
The $\underline{\text{only}}$ way that there would $\underline{\text{not be a solution}}$ is if the gradients were the same (parallel) and they did not coincide. If they did coincide (one on top of the other) then they would have an infinite number of shared points thus an infinite number of solutions.

$\textcolor{b l u e}{\implies \text{as there is a solution then they "ul("are 'consistent'.}}$

Observe that from the graph the two lines coincide at:

$\left(x , y\right) = \left(0 , - 10\right)$