ax+b=c

[Instructional Technologis]

ax+b=c When listed as Ax+By=C, or y-y1=m(x-x1) or mx+b=c, "c" is explained as a constant. However, I have not yet seen an explanation as to why in ax+b=c, y is missing and therefore what the c represents. In the other linear equations above it has been represented as a slope and as a y value depending on the multitude of interpretations reviewed. I know what all of the letters represent, but why is there an equation where y is missing when it is needed to determine the slope of the line. I have searched several web sites asking that question with no resultant explanation. Depending on the book being used to learn Algebra II each usually uses a different format. This is very confusing to students. I am currently re-writing a calculus book and formatting it based on learning theory - how the mind inputs, processes, stores and retrieves different kinds of knowledge from memory. Most learning theory is totally missing from mathematics texts, nor do instructors know the research about how the mind creates schema and organizes input for retrieval. The Algebra II for Dummies I am currently reviewing does not apply any theories on how a student learns, i.e., concepts and principles. All mathematical equations are principles: a set of conditions when applied result in a specific outcome. Each symbol in an equation is a concept; concepts are identified by their definitions; principles are identified by their proposition. They are learned differently. OK, I went a bit far but the question remains why is there no "y" in ax+b=c. In fact as I look it up, every source begins with Ax+By=C and doesn't even address ax+b=c. No wonder students have difficulty with math. I want to know "why" there is a lack of consistency in formatting and "why" the original equation is not addressed but altered in its form with no explanation.

Thanks,
Instructionl Technologist

 

[lev888]

Please post a link to the source.

 

[Dr.Peterson]

The equation ax + b = c is an entirely different kind of equation than the others you mention; having only one variable, it represents a line only if it is presented in a 2-dimensional context, in which case it is the vertical line x = (c-b)/a. (Actually, "mx+b=c" is exactly the same equation, so I'm supposing you meant "y = mx + b", or something like that.)

So, echoing what lev888 said, please show us the context in which you saw this equation written. I can't think of any setting where it would be presented as a form of equation, other then when first introducing linear equations in one variable (as a step before the general case, ax + b = cx + d, where the variable is on both sides). And this is far from any discussion of equations of lines, much less Algebra II.

The answer to your question will lie in the context -- why someone used that equation in the first place.

 

[Steven G]

I have not yet seen an explanation as to why in ax+b=c, y is missing and therefore what the c represents.
First note that ax+b=c is the same as ax=(c-b). Let's just say (c-b) = d.
Before I go on I want to point out in your equation ax+b=c, that c has no meaning at all as it can be changed to any number you want. For example ax+7 = 9 (so c=9) can also be written as ax = 2 (so c=2) or as ax + 11 = 13 or ax - 2 = 0, ....

So back to thinking of ax+b =c as ax=b. But dividing by a yields that x =b/a. That is x equals a constant, like in the equation x=5 or x= -2/3 or x=pi.
So in x = 7, x can NOT change! But y can! For example if I tell you that y=3 and I ask you what x is, you better say that x=7. Say if you I tell you that y = -2/3....x=7.

Now draw this line and tell me what the slope is. If you get the right answer then you will should know the answer to but why is there a linear equation where y is missing when it is needed to determine the slope of the line.

 

[Dr.Peterson]

ax+b=c ...
I know what all of the letters represent, but why is there an equation where y is missing when it is needed to determine the slope of the line. ...
the question remains why is there no "y" in ax+b=c. In fact as I look it up, every source begins with Ax+By=C and doesn't even address ax+b=c. ... I want to know "why" there is a lack of consistency in formatting and "why" the original equation is not addressed but altered in its form with no explanation.

I'm still hoping to give you a clear explanation, but in order to do that, I need to see just what was being said in the context where you found ax+b=c. (In communication, context is everything!) If you can provide a link to that source, or an image of the page, we can probably help you understand why it is said the way it is. (I tried searching for that equation in Algebra II for Dummies, and only got one hit, which didn't fit your description.)

I hope that will also help answer your larger question about "lack of consistency in formatting", by seeing how different contexts may require different details, or even why it's beneficial to students to see some variation in the way things are presented.

 

[HallsofIvy]

Another way of thinking about it is that "ax+ b= c" has a single value of x, \(\displaystyle \frac{c-b}{a}\), and, geometrically, corresponds to a single point on a one dimensional "number line". An equation like "ax+ by= c" is satisfied by infinitely many (x, y) pairs and, geometrically, corresponds to a line on a two dimensional xy- coordinate system.