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and change all the challenges to new `md` format.
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Oliver Eyton-Williams
2020-11-27 19:02:05 +01:00
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curriculum/challenges/english/10-coding-interview-prep/rosetta-code/cramers-rule.md
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@ -1,74 +1,57 @@
---
title: Cramer's rule
id: 59713da0a428c1a62d7db430
title: Cramer's rule
challengeType: 5
forumTopicId: 302239
---
## Description
<section id='description'>
In <a href="https://en.wikipedia.org/wiki/linear algebra" title="wp: linear algebra" target="_blank">linear algebra</a>, <a href="https://en.wikipedia.org/wiki/Cramer's rule" title="wp: Cramer's rule" target="_blank">Cramer's rule</a> is an explicit formula for the solution of a <a href="https://en.wikipedia.org/wiki/system of linear equations" title="wp: system of linear equations" target="_blank">system of linear equations</a> with as many equations as unknowns, valid whenever the system has a unique solution. It expresses the solution in terms of the determinants of the (square) coefficient matrix and of matrices obtained from it by replacing one column by the vector of right hand sides of the equations.
# --description--
In [linear algebra](<https://en.wikipedia.org/wiki/linear algebra> "wp: linear algebra"), [Cramer's rule](<https://en.wikipedia.org/wiki/Cramer's rule> "wp: Cramer's rule") is an explicit formula for the solution of a [system of linear equations](<https://en.wikipedia.org/wiki/system of linear equations> "wp: system of linear equations") with as many equations as unknowns, valid whenever the system has a unique solution. It expresses the solution in terms of the determinants of the (square) coefficient matrix and of matrices obtained from it by replacing one column by the vector of right hand sides of the equations.
Given
<big>
$\left\{\begin{matrix}a_1x + b_1y + c_1z&= {\color{red}d_1}\\a_2x + b_2y + c_2z&= {\color{red}d_2}\\a_3x + b_3y + c_3z&= {\color{red}d_3}\end{matrix}\right.$
</big>
$\\left\\{\\begin{matrix}a_1x + b_1y + c_1z&= {\\color{red}d_1}\\\\a_2x + b_2y + c_2z&= {\\color{red}d_2}\\\\a_3x + b_3y + c_3z&= {\\color{red}d_3}\\end{matrix}\\right.$
which in matrix format is
<big>
$\begin{bmatrix} a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3 \end{bmatrix}\begin{bmatrix} x \\ y \\ z \end{bmatrix}=\begin{bmatrix} {\color{red}d_1} \\ {\color{red}d_2} \\ {\color{red}d_3} \end{bmatrix}.$
</big>
$\\begin{bmatrix} a_1 & b_1 & c_1 \\\\ a_2 & b_2 & c_2 \\\\ a_3 & b_3 & c_3 \\end{bmatrix}\\begin{bmatrix} x \\\\ y \\\\ z \\end{bmatrix}=\\begin{bmatrix} {\\color{red}d_1} \\\\ {\\color{red}d_2} \\\\ {\\color{red}d_3} \\end{bmatrix}.$
Then the values of $x, y$ and $z$ can be found as follows:
<big>
$x = \frac{\begin{vmatrix} {\color{red}d_1} & b_1 & c_1 \\ {\color{red}d_2} & b_2 & c_2 \\ {\color{red}d_3} & b_3 & c_3 \end{vmatrix} } { \begin{vmatrix} a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3 \end{vmatrix}}, \quad y = \frac {\begin{vmatrix} a_1 & {\color{red}d_1} & c_1 \\ a_2 & {\color{red}d_2} & c_2 \\ a_3 & {\color{red}d_3} & c_3 \end{vmatrix}} {\begin{vmatrix} a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3 \end{vmatrix}}, \text{ and }z = \frac { \begin{vmatrix} a_1 & b_1 & {\color{red}d_1} \\ a_2 & b_2 & {\color{red}d_2} \\ a_3 & b_3 & {\color{red}d_3} \end{vmatrix}} {\begin{vmatrix} a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3 \end{vmatrix} }.$
</big>
</section>
## Instructions
<section id='instructions'>
$x = \\frac{\\begin{vmatrix} {\\color{red}d_1} & b_1 & c_1 \\\\ {\\color{red}d_2} & b_2 & c_2 \\\\ {\\color{red}d_3} & b_3 & c_3 \\end{vmatrix} } { \\begin{vmatrix} a_1 & b_1 & c_1 \\\\ a_2 & b_2 & c_2 \\\\ a_3 & b_3 & c_3 \\end{vmatrix}}, \\quad y = \\frac {\\begin{vmatrix} a_1 & {\\color{red}d_1} & c_1 \\\\ a_2 & {\\color{red}d_2} & c_2 \\\\ a_3 & {\\color{red}d_3} & c_3 \\end{vmatrix}} {\\begin{vmatrix} a_1 & b_1 & c_1 \\\\ a_2 & b_2 & c_2 \\\\ a_3 & b_3 & c_3 \\end{vmatrix}}, \\text{ and }z = \\frac { \\begin{vmatrix} a_1 & b_1 & {\\color{red}d_1} \\\\ a_2 & b_2 & {\\color{red}d_2} \\\\ a_3 & b_3 & {\\color{red}d_3} \\end{vmatrix}} {\\begin{vmatrix} a_1 & b_1 & c_1 \\\\ a_2 & b_2 & c_2 \\\\ a_3 & b_3 & c_3 \\end{vmatrix} }.$
# --instructions--
Given the following system of equations:
<big>
$\begin{cases}
2w-x+5y+z=-3 \\
3w+2x+2y-6z=-32 \\
w+3x+3y-z=-47 \\
5w-2x-3y+3z=49 \\
\end{cases}$
</big>
solve for <big>$w$, $x$, $y$</big> and <big>$z$</big>, using Cramer's rule.
</section>
## Tests
<section id='tests'>
$\\begin{cases} 2w-x+5y+z=-3 \\\\ 3w+2x+2y-6z=-32 \\\\ w+3x+3y-z=-47 \\\\ 5w-2x-3y+3z=49 \\\\ \\end{cases}$
```yml
tests:
- text: <code>cramersRule</code> should be a function.
testString: assert(typeof cramersRule === 'function');
- text: <code>cramersRule([[2, -1, 5, 1], [3, 2, 2, -6], [1, 3, 3, -1], [5, -2, -3, 3]], [-3, -32, -47, 49])</code> should return <code>[2, -12, -4, 1]</code>.
testString: assert.deepEqual(cramersRule(matrices[0], freeTerms[0]), answers[0]);
- text: <code>cramersRule([[3, 1, 1], [2, 2, 5], [1, -3, -4]], [3, -1, 2])</code> should return <code>[1, 1, -1]</code>.
testString: assert.deepEqual(cramersRule(matrices[1], freeTerms[1]), answers[1]);
solve for $w$, $x$, $y$ and $z$, using Cramer's rule.
```
# --hints--
</section>
## Challenge Seed
<section id='challengeSeed'>
<div id='js-seed'>
`cramersRule` should be a function.
```js
function cramersRule(matrix, freeTerms) {
return true;
}
assert(typeof cramersRule === 'function');
```
</div>
`cramersRule([[2, -1, 5, 1], [3, 2, 2, -6], [1, 3, 3, -1], [5, -2, -3, 3]], [-3, -32, -47, 49])` should return `[2, -12, -4, 1]`.
```js
assert.deepEqual(cramersRule(matrices[0], freeTerms[0]), answers[0]);
```
### After Test
<div id='js-teardown'>
`cramersRule([[3, 1, 1], [2, 2, 5], [1, -3, -4]], [3, -1, 2])` should return `[1, 1, -1]`.
```js
assert.deepEqual(cramersRule(matrices[1], freeTerms[1]), answers[1]);
```
# --seed--
## --after-user-code--
```js
const matrices = [
@ -89,13 +72,16 @@ const freeTerms = [[-3, -32, -47, 49], [3, -1, 2]];
const answers = [[2, -12, -4, 1], [1, 1, -1]];
```
</div>
## --seed-contents--
</section>
```js
function cramersRule(matrix, freeTerms) {
## Solution
<section id='solution'>
return true;
}
```
# --solutions--
```js
/**
@ -177,7 +163,4 @@ function detr(m) {
function clone(m) {
return m.map(a => a.slice());
}
```
</section>