Question

How do you tell if a system is consistent or inconsistent `2x + y = 7` and `x - 2y = 7`?

Answer 1

Consistent system of equations have at least one solution
Inconsistent system of equations have no solution.

we have equations
`2x + y = 7` ........equation `(1)`
`x - 2y =7` ..........equation `(2)`

multiplying equation `1` by `2` :
`( 2x + y = 7) xx 2`

the equation becomes:
`4x + 2y = 14`.............equation `(3)`
solving by elimination :

adding equations `2 and 3`:
`x -cancel (2y) =7`
`4x + cancel(2y) = 14`
`5x = 21`
`x=21/5` , substituting this value of `x` in equation 1 to obtain `y`:

`2x + y = 7`
`y = 7 - 2x`
`y = 7 -2 xx(21/5) = 7-42/5 = (35-42)/5 = -7/5`

the solution for the system of equations are :
`color(blue)x=21/5`
`color(blue)y=-7/5`

hence the equations are consistent as we have obtained a solution.