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#?

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

#x + y = 2 and#

#2x - 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 and# #..color(red)(Eqn.1)#

#2x - y =1# #..color(red)(Eqn.2)#

If you refer to the graph available with this solution, you can observe two distinct intersecting straight lines: one #color(blue)(Blue)# Line and one #color(red)(Red)# Line.

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

As you can observe, the intersection point has coordinates #(1,1)#

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

The solution set has single ordered pair #(1,1)#

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

Slope-Intercept Form is written as #y= mx+ b#

#m# is the Slope and #b# is the y_intercept

We can write #..color(red)(Eqn.1)# as

#y = -x+2# #..color(green)(Eqn.3)#

We can write #..color(red)(Eqn.2)# as

#y = 2x - 1# #..color(green)(Eqn.4)#

We observe that, using #..color(green)(Eqn.3)# and #..color(green)(Eqn.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.