Mixing
Standard form of a linear equation
$begingroup$
I am teaching a class on “intermediate algebra” and I noticed that there are two definitions for the standard form of a linear equation in two variables $x$ and $y$ in the textbook that I am using.
$1.$ A linear equation in two variables (here $x$ and $y$) can be written in the form $$Ax + By =C$$ where $A$, $B$ and $C$ are real numbers and $A$ and $B$ are not both $0$.
$2.$ $Ax+By = C$ where $A, B$ and $C$ are integers, $A geq 0$.
The first definition makes sense to me in that it is merely a non-trivial real-linear combination of $x$ and $y$ to create $C$ but the second definition seems to be awfully specific. It requires an integer-linear combination to form an integer $C$ and it also requires that $A$ be non-negative.
Now, perhaps insisting that $A, B$ and $C$ be integers follows from the fact that the book won’t be handling any real numbers that aren’t rational but I’m not at all sure why there is an insistence on $A$ being non-negative. Is there some form of advantage to the requirement that $A geq 0$ or is it perhaps just to make sure that a student is repeating the exact same process over and over again without needing to think it over or some other reason entirely?
Note: I know essentially nothing about math education and so this question may be very silly.
If I should add or remove a tag then feel free to let me know.
Thanks in advance!
algebra-precalculus education
asked Jan 24 at 22:06
user328442user328442
1,8901516
$endgroup$
add a comment |
$begingroup$
I am teaching a class on “intermediate algebra” and I noticed that there are two definitions for the standard form of a linear equation in two variables $x$ and $y$ in the textbook that I am using.
$1.$ A linear equation in two variables (here $x$ and $y$) can be written in the form $$Ax + By =C$$ where $A$, $B$ and $C$ are real numbers and $A$ and $B$ are not both $0$.
$2.$ $Ax+By = C$ where $A, B$ and $C$ are integers, $A geq 0$.
The first definition makes sense to me in that it is merely a non-trivial real-linear combination of $x$ and $y$ to create $C$ but the second definition seems to be awfully specific. It requires an integer-linear combination to form an integer $C$ and it also requires that $A$ be non-negative.
Now, perhaps insisting that $A, B$ and $C$ be integers follows from the fact that the book won’t be handling any real numbers that aren’t rational but I’m not at all sure why there is an insistence on $A$ being non-negative. Is there some form of advantage to the requirement that $A geq 0$ or is it perhaps just to make sure that a student is repeating the exact same process over and over again without needing to think it over or some other reason entirely?
Note: I know essentially nothing about math education and so this question may be very silly.
If I should add or remove a tag then feel free to let me know.
Thanks in advance!
algebra-precalculus education
asked Jan 24 at 22:06
user328442user328442
1,8901516
$endgroup$
1
$begingroup$
I see no reason for enforcing integers, even if the examples are only rational. Requiring $Age0$ is a fossil of the times when negative numbers were looked upon very suspiciously.
$endgroup$
– egreg
Jan 24 at 22:10
add a comment |
$begingroup$
I am teaching a class on “intermediate algebra” and I noticed that there are two definitions for the standard form of a linear equation in two variables $x$ and $y$ in the textbook that I am using.
$1.$ A linear equation in two variables (here $x$ and $y$) can be written in the form $$Ax + By =C$$ where $A$, $B$ and $C$ are real numbers and $A$ and $B$ are not both $0$.
$2.$ $Ax+By = C$ where $A, B$ and $C$ are integers, $A geq 0$.
The first definition makes sense to me in that it is merely a non-trivial real-linear combination of $x$ and $y$ to create $C$ but the second definition seems to be awfully specific. It requires an integer-linear combination to form an integer $C$ and it also requires that $A$ be non-negative.
Now, perhaps insisting that $A, B$ and $C$ be integers follows from the fact that the book won’t be handling any real numbers that aren’t rational but I’m not at all sure why there is an insistence on $A$ being non-negative. Is there some form of advantage to the requirement that $A geq 0$ or is it perhaps just to make sure that a student is repeating the exact same process over and over again without needing to think it over or some other reason entirely?
Note: I know essentially nothing about math education and so this question may be very silly.
If I should add or remove a tag then feel free to let me know.
Thanks in advance!
algebra-precalculus education
asked Jan 24 at 22:06
user328442user328442
1,8901516
$endgroup$
I am teaching a class on “intermediate algebra” and I noticed that there are two definitions for the standard form of a linear equation in two variables $x$ and $y$ in the textbook that I am using.
$1.$ A linear equation in two variables (here $x$ and $y$) can be written in the form $$Ax + By =C$$ where $A$, $B$ and $C$ are real numbers and $A$ and $B$ are not both $0$.
$2.$ $Ax+By = C$ where $A, B$ and $C$ are integers, $A geq 0$.
The first definition makes sense to me in that it is merely a non-trivial real-linear combination of $x$ and $y$ to create $C$ but the second definition seems to be awfully specific. It requires an integer-linear combination to form an integer $C$ and it also requires that $A$ be non-negative.
Now, perhaps insisting that $A, B$ and $C$ be integers follows from the fact that the book won’t be handling any real numbers that aren’t rational but I’m not at all sure why there is an insistence on $A$ being non-negative. Is there some form of advantage to the requirement that $A geq 0$ or is it perhaps just to make sure that a student is repeating the exact same process over and over again without needing to think it over or some other reason entirely?
Note: I know essentially nothing about math education and so this question may be very silly.
If I should add or remove a tag then feel free to let me know.
Thanks in advance!
algebra-precalculus education
algebra-precalculus education
asked Jan 24 at 22:06
user328442user328442
1,8901516
asked Jan 24 at 22:06
user328442user328442
1,8901516
asked Jan 24 at 22:06
user328442user328442
1,8901516
asked Jan 24 at 22:06
user328442user328442
1,8901516
asked Jan 24 at 22:06
user328442user328442
1,8901516
1,8901516
1
$begingroup$
I see no reason for enforcing integers, even if the examples are only rational. Requiring $Age0$ is a fossil of the times when negative numbers were looked upon very suspiciously.
$endgroup$
– egreg
Jan 24 at 22:10
add a comment |
1
$begingroup$
I see no reason for enforcing integers, even if the examples are only rational. Requiring $Age0$ is a fossil of the times when negative numbers were looked upon very suspiciously.
$endgroup$
– egreg
Jan 24 at 22:10
1
1
$begingroup$
I see no reason for enforcing integers, even if the examples are only rational. Requiring $Age0$ is a fossil of the times when negative numbers were looked upon very suspiciously.
$endgroup$
– egreg
Jan 24 at 22:10
$begingroup$
I see no reason for enforcing integers, even if the examples are only rational. Requiring $Age0$ is a fossil of the times when negative numbers were looked upon very suspiciously.
$endgroup$
– egreg
Jan 24 at 22:10
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
I can think of two reasons. One is that by requiring $A$ to be positive, slightly less ink is used in the equation, or at least on the left side, and so this might be considered "simpler."
The other is that often students are asked to put answers in a certain form just to make it easier for the teacher to grade. (Or nowadays, with the online homework servers, so that the computer doesn't mark a correct answer wrong.)
answered Jan 24 at 22:24
B. GoddardB. Goddard
19.6k21442
$endgroup$
add a comment |
Your Answer
StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");
StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});
function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});
}
});
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3086414%2fstandard-form-of-a-linear-equation%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
I can think of two reasons. One is that by requiring $A$ to be positive, slightly less ink is used in the equation, or at least on the left side, and so this might be considered "simpler."
The other is that often students are asked to put answers in a certain form just to make it easier for the teacher to grade. (Or nowadays, with the online homework servers, so that the computer doesn't mark a correct answer wrong.)
answered Jan 24 at 22:24
B. GoddardB. Goddard
19.6k21442
$endgroup$
add a comment |
$begingroup$
I can think of two reasons. One is that by requiring $A$ to be positive, slightly less ink is used in the equation, or at least on the left side, and so this might be considered "simpler."
The other is that often students are asked to put answers in a certain form just to make it easier for the teacher to grade. (Or nowadays, with the online homework servers, so that the computer doesn't mark a correct answer wrong.)
answered Jan 24 at 22:24
B. GoddardB. Goddard
19.6k21442
$endgroup$
add a comment |
$begingroup$
I can think of two reasons. One is that by requiring $A$ to be positive, slightly less ink is used in the equation, or at least on the left side, and so this might be considered "simpler."
The other is that often students are asked to put answers in a certain form just to make it easier for the teacher to grade. (Or nowadays, with the online homework servers, so that the computer doesn't mark a correct answer wrong.)
answered Jan 24 at 22:24
B. GoddardB. Goddard
19.6k21442
$endgroup$
I can think of two reasons. One is that by requiring $A$ to be positive, slightly less ink is used in the equation, or at least on the left side, and so this might be considered "simpler."
The other is that often students are asked to put answers in a certain form just to make it easier for the teacher to grade. (Or nowadays, with the online homework servers, so that the computer doesn't mark a correct answer wrong.)
answered Jan 24 at 22:24
B. GoddardB. Goddard
19.6k21442
answered Jan 24 at 22:24
B. GoddardB. Goddard
19.6k21442
answered Jan 24 at 22:24
B. GoddardB. Goddard
19.6k21442
answered Jan 24 at 22:24
B. GoddardB. Goddard
19.6k21442
19.6k21442
add a comment |
add a comment |
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3086414%2fstandard-form-of-a-linear-equation%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
1
$begingroup$
I see no reason for enforcing integers, even if the examples are only rational. Requiring $Age0$ is a fossil of the times when negative numbers were looked upon very suspiciously.
$endgroup$
– egreg
Jan 24 at 22:10