# How do you solve the system of equations by graphing and then classify the system as consistent or inconsistent x+y=2 and 2x-y=1?

Dec 5, 2017

Please refer to the graph and the explanation.

#### Explanation:

We are given systems of two linear equations in two variables:

$x + y = 2 \mathmr{and}$

$2 x - y = 1$

These can be visually represented by simultaneously graphing both the equations.

The system can be Consistent or Inconsistent and the equations in the system can either be Dependent or Independent.

A system which has No Solutions are said to be Inconsistent.

A system with one or more solutions are called Consistent, having either one solution or an infinite number of solutions.

We are given systems of two linear equations in two variables:

$x + y = 2 \mathmr{and}$ $. . \textcolor{red}{E q n .1}$

$2 x - y = 1$ $. . \textcolor{red}{E q n .2}$

If you refer to the graph available with this solution, you can observe two distinct intersecting straight lines: one $\textcolor{b l u e}{B l u e}$ Line and one $\textcolor{red}{R e d}$ Line.

We get a pair of $\left(x , y\right)$ which is the single unique solution for the system of equations.

As you can observe, the intersection point has coordinates $\left(1 , 1\right)$

Our system of equations is therefore a Consistent System of Independent Equations.

The solution set has single ordered pair $\left(1 , 1\right)$

We will write our equations in the Slope-Intercept Form:

Slope-Intercept Form is written as $y = m x + b$

$m$ is the Slope and $b$ is the y_intercept

We can write $. . \textcolor{red}{E q n .1}$ as

$y = - x + 2$ $. . \textcolor{g r e e n}{E q n .3}$

We can write $. . \textcolor{red}{E q n .2}$ as

$y = 2 x - 1$ $. . \textcolor{g r e e n}{E q n .4}$

We observe that, using $. . \textcolor{g r e e n}{E q n .3}$ and $. . \textcolor{g r e e n}{E q n .4}$

The Slope is different from each equation.

The system has One Solution and therefore is a Consistent System.

The equations are also Independent, as each equation is describing a different straight line.

I hope this helps.