Question

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`?

Answer 1

Please refer to the graph and the explanation.

Explanation:

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.