Linear equations, how many solutions does it really have?
My math book says, a Linear equation has exactly one solution. Because $ax + b = 0$; $x =-\frac{b}{a}$. But I've solved many linear equations with multiple solutions before. (I'm not very good in math. Need help...)
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$\begingroup$If we assume that all linear equations have the form:
$$ ax + b= 0 $$
(which is completely valid and should be how we view linear equations)
then linear equations have either 1, 0, or infinite solutions. It's quite simple if $a \neq 0$ then they have exactly one solution: $x = -\frac{b}{a}$.
On the other hand, if $a = 0$ then if $b = 0$ we have infinite solutions (any value of $x$ solves $0x + 0 = 0$) and if $a = 0$ and $b \neq 0$ then there are no solutions (there is no value of $x$ that makes $0x + 1 = 0$, for example).
$\endgroup$ $\begingroup$If we have $2$ unknowns, then the linear system
$$a_1 x_1 + a_2 x_2 = b$$
has, in general, infinitely many solutions. Why is that? Assuming that $a_1 \neq 0$, we write
$$x_1 = \frac{b}{a_1} - \left(\frac{a_2}{a_1}\right) x_2$$
Let $x_2 = \gamma$, where $\gamma \in \mathbb{R}$. Then, the solution set is a line parameterized as follows
$$\begin{bmatrix}x_1\\ x_2\end{bmatrix} = \begin{bmatrix} \frac{b}{a_1}\\ 0\end{bmatrix} + \gamma \begin{bmatrix} - \frac{a_2}{a_1}\\ 1\end{bmatrix}$$
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