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

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

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Answer 1 · 2015-05-20T08:09:08

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.

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

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