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A few weeks ago, my son came to me asking why it was that the automated math homework program the school used was telling him the sum of two linear expressions wasn't always a linear expression. At this point, our questions are for our own interest piqued from the homework problem. As someone who deals mostly with equations, and not expressions, it's taken a few weeks time (we're busy) to figure out due to a lack of my familiarity with how things are defined, so let me share some definitions before I ask my questions.

Expression

"Expression is a mathematical phrase which combines, numbers, variables and operators to show the value of something." It does not contain an equals sign. Difference Between Expression and Equation

They like to exchange the word evaluate for equal in this context for when you plug in the value of a variable and solve the problem.

For Example:

$5x + 4$ is an expression, but $0 = 5x + 4$ is not.

Linear

Rather than having a graphical/geometric definition about a line or of being of a certain equation form, the linear in this instance is saying that the power of each variable in the expression is $1$.

For example:

$5x$, $5x+5y$, $5x+2$ are all linear. $5x^2$ is not linear.

The Questions:

The solution to the answer implies the following. Let's assume you have two a linear expressions, where one expression has the opposite variable value from the other expression (for example: $5x+2$ and $-5x+2$), then the addition of those two expressions is a constant, and not a linear expression, since you don't have a variable with a power of $1$.

  1. Why would an answer to the previous example of the addition of the two expressions $5x+2$ and $-5x+2$ not be $0x+2$ rather than just $2$? Would $0x+2$ still be considered a linear expression given the above definitions? When we would deal with linear functions later on, if you have $f(x) = 2$, then you have an implied $0x$ in there. Why is this considered different?

  2. Are these definitions in common usage, or is this something generated by particular author / publisher? I believe it's supposed to relate back to the Common Core standard for grades 6-8 on Expressions and Equations, but I wasn't able to figure out how.

  3. Where do these definitions come from with regards to what branch of mathematics? I've mostly be scavenging them off the web.

  4. Where is this form used other than in the teaching of mathematics? Where can I go to learn more about how it's used, and why it exists?

    I'd like to be able to either explain to my child where, how, and why it's used in this manner. If it's just another one of those building steps with an extra twist that are used in school to get to the next topic (linear equations) and that once he gets beyond this year, he can basically ignore for usage in doing the rest of mathematics that's fine also.

  5. It seems like there is an implied equality in this evaluation. Even in the different places where I see it utilized in the children's school books, I see them make use of the equals sign, once they talk about evaluating the expression. What is the difference that is attempting to be taught with the distinction between evaluation and equality in this context?

Additions based on comments:

The computerized system itself provided a statement, where you could change the answer from it being valid or not. It was of the form paraphrased "Your friend says the sum of two linear expressions is always a linear expression. Is he Right?" You were then given a statement where you could change the wording a statement that was like the following, again paraphrased. "If you add two linear expression where one expression has the opposite [variable | constant] value, then the result of adding them together [is|is not] a linear expression.

Here's an example where it would indicate that the addition of these two expressions is not a linear expression.

  • Expression 1: $4x+2$
  • Expression 2: $-4x+2$
  • $(4x+2) + (-4x+2) \implies 2$
  • $2$ is a constant, and thus not a linear expression.

I've never seen this particular definition for linear used anywhere else. I'm more familiar with the term linear as the comment by JMoravitz described using the term affine or in the algebraic equation forms $y=mx+b$ and its related forms. My questions relate to where does this definition of linear come from and where can I learn more about it?

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  • $\begingroup$ Can you provide an example of two "linear expressions" such that the system did not believe that their sum was linear? Automated systems can be deeply stupid, of course. I'd have said that "linear expression" here just meant "$ax+b$ for some $a,b\in \mathbb R$". Maybe the system demands $a\neq 0$? $\endgroup$ – lulu Oct 15 at 16:33
  • $\begingroup$ If a linear expression is defined as "the power of each variable is 1", then any constant is (vacuously) linear since there are no variables. However, I don't think this is good definition, for instance $y^2-y^2+x$ would not be linear under this definition despite being equivalent to $x$. $\endgroup$ – 79037662 Oct 15 at 16:34
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    $\begingroup$ As an aside, because of the confusion between linear in the "the graph of a line" sense and linear in the linear algebra sense, you will see the graph of a line meaning be called instead "affine" to avoid further confusion at higher levels. Unfortunately, lower level textbooks and the general public have not yet caught on to the convention. $\endgroup$ – JMoravitz Oct 15 at 16:37
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    $\begingroup$ @Jean-ClaudeArbaut In English too! Though, as I say, elementary texts tend to be pretty careless regarding the distinction. $\endgroup$ – lulu Oct 15 at 16:38
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    $\begingroup$ As an additional aside, in terms of linear from the linear algebra sense, $f(x)=5x+2$ is not linear (though it is affine) since it does not satisfy the requirement that $f(\alpha x_1+\beta x_2)=\alpha f(x_1)+\beta f(x_2)$, for instance $f(0x)=2\neq 0 = 0f(x)$ $\endgroup$ – JMoravitz Oct 15 at 16:44
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Short answer: the solution provided is wrong. Linearity should not be characterized by "the value of power to which $x$ is raised".

A precise definition depends on the context - elementary school, algebra I, linear algebra in college. Your argument about lines in the plane is the right one for grade 6.

With any reasonable definition, the sum of linear things will be linear.

For a discussion of the equals sign, this may help:

What exactly is an equation?

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