Laplace Transform Simultaneous Equations
$begingroup$
If I have the matrix down below how can I get rid of I1 so I can have everything in terms of I2?
$
begin{pmatrix}
(10+s) & (-s-6) \
(-s-6) & (s+dfrac{4}{s}+6) \
end{pmatrix}
$
$ begin{pmatrix}
I1\
I2\
end{pmatrix}
= begin{pmatrix}
dfrac{6}{s+3}\
dfrac{-6}{s}-1\
end{pmatrix} $
This is Laplace Transform so I can have everything in terms of $s$.
laplace-transform
asked Jan 21 at 0:48
user154844user154844
53
$endgroup$
add a comment |
$begingroup$
If I have the matrix down below how can I get rid of I1 so I can have everything in terms of I2?
$
begin{pmatrix}
(10+s) & (-s-6) \
(-s-6) & (s+dfrac{4}{s}+6) \
end{pmatrix}
$
$ begin{pmatrix}
I1\
I2\
end{pmatrix}
= begin{pmatrix}
dfrac{6}{s+3}\
dfrac{-6}{s}-1\
end{pmatrix} $
This is Laplace Transform so I can have everything in terms of $s$.
laplace-transform
asked Jan 21 at 0:48
user154844user154844
53
$endgroup$
add a comment |
$begingroup$
If I have the matrix down below how can I get rid of I1 so I can have everything in terms of I2?
$
begin{pmatrix}
(10+s) & (-s-6) \
(-s-6) & (s+dfrac{4}{s}+6) \
end{pmatrix}
$
$ begin{pmatrix}
I1\
I2\
end{pmatrix}
= begin{pmatrix}
dfrac{6}{s+3}\
dfrac{-6}{s}-1\
end{pmatrix} $
This is Laplace Transform so I can have everything in terms of $s$.
laplace-transform
asked Jan 21 at 0:48
user154844user154844
53
$endgroup$
If I have the matrix down below how can I get rid of I1 so I can have everything in terms of I2?
$
begin{pmatrix}
(10+s) & (-s-6) \
(-s-6) & (s+dfrac{4}{s}+6) \
end{pmatrix}
$
$ begin{pmatrix}
I1\
I2\
end{pmatrix}
= begin{pmatrix}
dfrac{6}{s+3}\
dfrac{-6}{s}-1\
end{pmatrix} $
This is Laplace Transform so I can have everything in terms of $s$.
laplace-transform
laplace-transform
asked Jan 21 at 0:48
user154844user154844
53
asked Jan 21 at 0:48
user154844user154844
53
asked Jan 21 at 0:48
user154844user154844
53
asked Jan 21 at 0:48
user154844user154844
53
asked Jan 21 at 0:48
user154844user154844
53
53
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
Reducing by row, observing that writing $a=s+6$, $b=frac{6}{s+3}$ and $c=-frac{6}{s}-1$ the augmented matrix becomes
$$
left(
begin{array}{cc|c}
4+a & -a & b\
-a & frac{4}{s}+a & c
end{array}
right)xrightarrow{R_2leftrightarrow R_2+R_1}
left(
begin{array}{cc|c}
4+a & -a & b\
4 & frac{4}{s} & b+c
end{array}
right)\
xrightarrow{R_1leftrightarrow frac{R_1}{4+a}}
left(
begin{array}{cc|c}
1 & -frac{a}{4+a} & frac{b}{4+a}\
4 & frac{4}{s} & b+c
end{array}
right)xrightarrow{R_2leftrightarrow R_2-4R_1}left(
begin{array}{cc|c}
1 & -frac{a}{4+a} & frac{b}{4+a}\
0 & frac{4}{s} + frac{4a}{4+a} & b+c-frac{4b}{4+a}
end{array}
right)
$$
So you have
begin{align*}
I_2&=frac{b+c-frac{4b}{4+a}}{frac{4}{s} + frac{4a}{4+a} }\
I_1&=frac{b}{4+a}+frac{a}{4+a}I_2
end{align*}
answered Jan 22 at 21:41
alexjoalexjo
12.5k1430
$endgroup$
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%2f3081350%2flaplace-transform-simultaneous-equations%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
$begingroup$
Reducing by row, observing that writing $a=s+6$, $b=frac{6}{s+3}$ and $c=-frac{6}{s}-1$ the augmented matrix becomes
$$
left(
begin{array}{cc|c}
4+a & -a & b\
-a & frac{4}{s}+a & c
end{array}
right)xrightarrow{R_2leftrightarrow R_2+R_1}
left(
begin{array}{cc|c}
4+a & -a & b\
4 & frac{4}{s} & b+c
end{array}
right)\
xrightarrow{R_1leftrightarrow frac{R_1}{4+a}}
left(
begin{array}{cc|c}
1 & -frac{a}{4+a} & frac{b}{4+a}\
4 & frac{4}{s} & b+c
end{array}
right)xrightarrow{R_2leftrightarrow R_2-4R_1}left(
begin{array}{cc|c}
1 & -frac{a}{4+a} & frac{b}{4+a}\
0 & frac{4}{s} + frac{4a}{4+a} & b+c-frac{4b}{4+a}
end{array}
right)
$$
So you have
begin{align*}
I_2&=frac{b+c-frac{4b}{4+a}}{frac{4}{s} + frac{4a}{4+a} }\
I_1&=frac{b}{4+a}+frac{a}{4+a}I_2
end{align*}
answered Jan 22 at 21:41
alexjoalexjo
12.5k1430
$endgroup$
add a comment |
$begingroup$
Reducing by row, observing that writing $a=s+6$, $b=frac{6}{s+3}$ and $c=-frac{6}{s}-1$ the augmented matrix becomes
$$
left(
begin{array}{cc|c}
4+a & -a & b\
-a & frac{4}{s}+a & c
end{array}
right)xrightarrow{R_2leftrightarrow R_2+R_1}
left(
begin{array}{cc|c}
4+a & -a & b\
4 & frac{4}{s} & b+c
end{array}
right)\
xrightarrow{R_1leftrightarrow frac{R_1}{4+a}}
left(
begin{array}{cc|c}
1 & -frac{a}{4+a} & frac{b}{4+a}\
4 & frac{4}{s} & b+c
end{array}
right)xrightarrow{R_2leftrightarrow R_2-4R_1}left(
begin{array}{cc|c}
1 & -frac{a}{4+a} & frac{b}{4+a}\
0 & frac{4}{s} + frac{4a}{4+a} & b+c-frac{4b}{4+a}
end{array}
right)
$$
So you have
begin{align*}
I_2&=frac{b+c-frac{4b}{4+a}}{frac{4}{s} + frac{4a}{4+a} }\
I_1&=frac{b}{4+a}+frac{a}{4+a}I_2
end{align*}
answered Jan 22 at 21:41
alexjoalexjo
12.5k1430
$endgroup$
add a comment |
$begingroup$
Reducing by row, observing that writing $a=s+6$, $b=frac{6}{s+3}$ and $c=-frac{6}{s}-1$ the augmented matrix becomes
$$
left(
begin{array}{cc|c}
4+a & -a & b\
-a & frac{4}{s}+a & c
end{array}
right)xrightarrow{R_2leftrightarrow R_2+R_1}
left(
begin{array}{cc|c}
4+a & -a & b\
4 & frac{4}{s} & b+c
end{array}
right)\
xrightarrow{R_1leftrightarrow frac{R_1}{4+a}}
left(
begin{array}{cc|c}
1 & -frac{a}{4+a} & frac{b}{4+a}\
4 & frac{4}{s} & b+c
end{array}
right)xrightarrow{R_2leftrightarrow R_2-4R_1}left(
begin{array}{cc|c}
1 & -frac{a}{4+a} & frac{b}{4+a}\
0 & frac{4}{s} + frac{4a}{4+a} & b+c-frac{4b}{4+a}
end{array}
right)
$$
So you have
begin{align*}
I_2&=frac{b+c-frac{4b}{4+a}}{frac{4}{s} + frac{4a}{4+a} }\
I_1&=frac{b}{4+a}+frac{a}{4+a}I_2
end{align*}
answered Jan 22 at 21:41
alexjoalexjo
12.5k1430
$endgroup$
Reducing by row, observing that writing $a=s+6$, $b=frac{6}{s+3}$ and $c=-frac{6}{s}-1$ the augmented matrix becomes
$$
left(
begin{array}{cc|c}
4+a & -a & b\
-a & frac{4}{s}+a & c
end{array}
right)xrightarrow{R_2leftrightarrow R_2+R_1}
left(
begin{array}{cc|c}
4+a & -a & b\
4 & frac{4}{s} & b+c
end{array}
right)\
xrightarrow{R_1leftrightarrow frac{R_1}{4+a}}
left(
begin{array}{cc|c}
1 & -frac{a}{4+a} & frac{b}{4+a}\
4 & frac{4}{s} & b+c
end{array}
right)xrightarrow{R_2leftrightarrow R_2-4R_1}left(
begin{array}{cc|c}
1 & -frac{a}{4+a} & frac{b}{4+a}\
0 & frac{4}{s} + frac{4a}{4+a} & b+c-frac{4b}{4+a}
end{array}
right)
$$
So you have
begin{align*}
I_2&=frac{b+c-frac{4b}{4+a}}{frac{4}{s} + frac{4a}{4+a} }\
I_1&=frac{b}{4+a}+frac{a}{4+a}I_2
end{align*}
answered Jan 22 at 21:41
alexjoalexjo
12.5k1430
edited Jan 22 at 22:40
answered Jan 22 at 21:41
alexjoalexjo
12.5k1430
answered Jan 22 at 21:41
alexjoalexjo
12.5k1430
answered Jan 22 at 21:41
alexjoalexjo
12.5k1430
12.5k1430
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%2f3081350%2flaplace-transform-simultaneous-equations%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
