I suspect that two issues may be partly underlying your intuition:

• First, the equation $y = mx + c$ gives $y$ as an explicit function of $x$, whereas $ax + by + c = 0$ only gives a relation between $x$ and $y$. This is a meaningful distinction, one worth making. However, the correct words for making this distinction are not "equation of a line" vs. "linear equation" (which mean exactly the same thing) but rather "function" vs. "relation". As other commenters have pointed out, the format $y = mx + c$ only describes non-vertical lines (which are functions), whereas $ax + by + c = 0$ also includes vertical lines (which are not functions).
• Second, in the equation $y = mx + c$ the parameters $m$ and $c$ are uniquely determined by the non-vertical line. That is: given any non-vertical line, there is a unique pair of numbers $(m, c)$ with the property that $y = mx + c$ describes the line; and conversely, given any pair of numbers $(m, c)$ there is a unique line given by the equation $y = mx + c$. In marked contrast, the parameters $a, b, c$ in the equation $ax + by + c = 0$ are not uniquely determined; if $ax + by + c = 0$ is an equation for some line $\ell$, then for any non-zero value $k$, the equation $kax + kby + kc = 0$ describes the same line. I suspect this may be what your intuition is recognizing when you say that $y = mx + b$ is "the address" of the line; it is a unique identifier of the line, not just one of many possible descriptions of it.

Having said all of this, it is definitely the case that the textbook is not wrong. There is indeed a distinction to be made between the two forms of the equation, but it is not the distinction you think it is.

Finally, I want to commend you for asking this question. Some of the comments have been condescending and even derogatory, and I think that's not warranted. It is hard to admit you don't completely understand something, especially when you are preparing to be a teacher. The whole point of this site is to answer questions about teaching math; if novices can't ask questions here, where else could they go?

I suspect that two issues may be partly underlying your intuition:

• First, the equation $y = mx + c$ gives $y$ as an explicit function of $x$, whereas $ax + by + c = 0$ only gives a relation between $x$ and $y$. This is a meaningful distinction, one worth making. However, the correct words for making this distinction are not "equation of a line" vs. "linear equation" (which mean exactly the same thing) but rather "function" vs. "relation". As other commenters have pointed out, the format $y = mx + c$ only describes non-vertical lines (which are functions), whereas $ax + by + c = 0$ also includes vertical lines (which are not functions). Another way of describing this distinction is that $y=mx + c$ defines $y$ explicitly in terms of $x$, whereas $ax + by + c = 0$ defines $y$ implicitly in terms of $x$ (assuming $b \ne 0$).
• Second, in the equation $y = mx + c$ the parameters $m$ and $c$ are uniquely determined by the non-vertical line. That is: given any non-vertical line, there is a unique pair of numbers $(m, c)$ with the property that $y = mx + c$ describes the line; and conversely, given any pair of numbers $(m, c)$ there is a unique line given by the equation $y = mx + c$. In marked contrast, the parameters $a, b, c$ in the equation $ax + by + c = 0$ are not uniquely determined; if $ax + by + c = 0$ is an equation for some line $\ell$, then for any non-zero value $k$, the equation $kax + kby + kc = 0$ describes the same line. I suspect this may be what your intuition is recognizing when you say that $y = mx + b$ is "the address" of the line; it is a unique identifier of the line, not just one of many possible descriptions of it.

Having said all of this, it is definitely the case that the textbook is not wrong. There is indeed a distinction to be made between the two forms of the equation, but it is not the distinction you think it is.

Finally, I want to commend you for asking this question. Some of the comments have been condescending and even derogatory, and I think that's not warranted. It is hard to admit you don't completely understand something, especially when you are preparing to be a teacher. The whole point of this site is to answer questions about teaching math; if novices can't ask questions here, where else could they go?

I suspect that two issues may be partly underlying your intuition:

• First, the equation $y = mx + c$ gives $y$ as an explicit function of $x$, whereas $ax + by + c = 0$ only gives a relation between $x$ and $y$. This is a meaningful distinction, one worth making. However, the correct words for making this distinction are not "equation of a line" vs. "linear equation" (which mean exactly the same thing) but rather "function" vs. "relation". As other commenters have pointed out, the format $y = mx + c$ only describes non-vertical lines (which are functions), whereas $ax + by + c = 0$ also includes vertical lines (which are not functions).
• Second, in the equation $y = mx + c$ the parameters $m$ and $c$ are uniquely determined by the non-vertical line. That is: given any non-vertical line, there is a unique pair of numbers $(m, c)$ with the property that $y = mx + c$ describes the line; and conversely, given any pair of numbers $(m, c)$ there is a unique line given by the equation $y = mx + c$. In marked contrast, the parameters $a, b, c$ in the equation $ax + by + c = 0$ are not uniquely determined; if $ax + by + c = 0$ is an equation for some line $\ell$, then for any non-zero value $k$, the equation $ka + kb + kc = 0$ describes the same line. I suspect this may be what your intuition is recognizing when you say that $y = mx + b$ is "the address" of the line; it is a unique identifier of the line, not just one of many possible descriptions of it.

Having said all of this, it is definitely the case that the textbook is not wrong. There is indeed a distinction to be made between the two forms of the equation, but it is not the distinction you think it is.

Finally, I want to commend you for asking this question. Some of the comments have been condescending and even derogatory, and I think that's not warranted. It is hard to admit you don't completely understand something, especially when you are preparing to be a teacher. The whole point of this site is to answer questions about teaching math; if novices can't ask questions here, where else could they go?

I suspect that two issues may be partly underlying your intuition:

• First, the equation $y = mx + c$ gives $y$ as an explicit function of $x$, whereas $ax + by + c = 0$ only gives a relation between $x$ and $y$. This is a meaningful distinction, one worth making. However, the correct words for making this distinction are not "equation of a line" vs. "linear equation" (which mean exactly the same thing) but rather "function" vs. "relation". As other commenters have pointed out, the format $y = mx + c$ only describes non-vertical lines (which are functions), whereas $ax + by + c = 0$ also includes vertical lines (which are not functions).
• Second, in the equation $y = mx + c$ the parameters $m$ and $c$ are uniquely determined by the non-vertical line. That is: given any non-vertical line, there is a unique pair of numbers $(m, c)$ with the property that $y = mx + c$ describes the line; and conversely, given any pair of numbers $(m, c)$ there is a unique line given by the equation $y = mx + c$. In marked contrast, the parameters $a, b, c$ in the equation $ax + by + c = 0$ are not uniquely determined; if $ax + by + c = 0$ is an equation for some line $\ell$, then for any non-zero value $k$, the equation $kax + kby + kc = 0$ describes the same line. I suspect this may be what your intuition is recognizing when you say that $y = mx + b$ is "the address" of the line; it is a unique identifier of the line, not just one of many possible descriptions of it.

Having said all of this, it is definitely the case that the textbook is not wrong. There is indeed a distinction to be made between the two forms of the equation, but it is not the distinction you think it is.

Finally, I want to commend you for asking this question. Some of the comments have been condescending and even derogatory, and I think that's not warranted. It is hard to admit you don't completely understand something, especially when you are preparing to be a teacher. The whole point of this site is to answer questions about teaching math; if novices can't ask questions here, where else could they go?