Linear equation with no solutions (parallel lines) [closed]

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Suppose I have two equations

-x + y = 0
-x + y = -2

Suppose I don't know geometry, I don't know slopes, suppose I have just started reading algebraic equations .

after solving those above equation .. I will get stuck to

-2 = 0

answer will be those equations don't have solutions...

But if I ask why? Why cant they have solutions. what will be the answer in terms of algebra.

I know -2 is not equal to 0 but those are mathematical values, but what is answer in terms of algebraic definition or anything?

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3 Answers

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Assume the reflexive property, x=x.$$ $$ y=x-2 $$ $$ y=x $$ $$ The reflexive property also says y=y. $$ $$ Therefore, x=x-2. $$ $$ This contradicts the reflexive property, x=x.$$ $$ For any solution to exist, it would have to contradict the reflexive property. But the reflexive property is a fundamental algebraic axiom, ergo no solution can exist.

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If you choose to not accept slopes / any geometric construction, then the best that I can think of in pure algebra is the number of common solutions between the lines.

1. If two lines are parallel, then they will never intersect (that is, they will have $0$ solutions in common).

2. If both the lines are the same line, then they will have infinite solutions in common.

3. If the lines are not parallel, then they will have $1$ unique solution in common

The number of solutions of a system of equations can be computed by writing the equation as a matrix equation

$$ A x = b $$

where $A$ is your coefficient matrix, $x$ is your vector of variables, and $b$ is your constants vector.

For this problem,

$$ -x + y = 0 \\ -x +y = -2 \\ \\ \begin{bmatrix}-1 & +1 \\ -1 &+1\end{bmatrix} \begin{bmatrix} x \\y \end{bmatrix} = \begin{bmatrix} 0 \\ 2 \end{bmatrix} $$

Hence,

$$ A = \begin{bmatrix}-1 & +1 \\ -1 &+1\end{bmatrix} \\ x =\begin{bmatrix} x \\y \end{bmatrix} \\ b = \begin{bmatrix} 0 \\ 2 \end{bmatrix} $$

The solution to the system $Ax = b$ is $x = A^{-1}b$. Hence, A must be invertible for solutions to exist

In our case, the determinant of $A$, $|A| = 0$ since it has the same row $\begin{pmatrix} -1 & +1 \end{pmatrix}$, hence the determinant will be $0$ and the matrix $A$ is non-invertible.

Since the matrix $A$ is non-invertible, but we need to achieve some non-zero value on the other side, the system of equations has no solution, and hence the lines are parallel

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There is no solution. This means there are no coordinates $(x,y)$ that satisfy both equations at the same time. This means there is is no point $P=(x,y)$ the can lie on both lines at the same time. Hence, the lines do not intersect. By definition the lines are parallel then.

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