1rBTX 03. EQUACIONS I INEQUACIONS/1.3.3 Sistemes 1.3.3.15Q ReduccióAleatori Determina les solucions de: «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mfenced mathcolor=¨#003300¨ open=¨{¨ close=¨}¨»«mtable»«mtr»«mtd»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»e«/mi»«mn mathvariant=¨bold¨»1«/mn»«/mtd»«/mtr»«mtr»«mtd»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»e«/mi»«mn mathvariant=¨bold¨»2«/mn»«/mtd»«/mtr»«/mtable»«/mfenced»«/mstyle»«/math»

Format de la resposta: Si el sistema és

compatible determinat: escriu la solució: {-21,11/15}
compatible indeterminat: escriu 1
incompatible, escriu 2

]]> 1.0000000 0.3333333 0 0 #sol <question><wirisCasSession><![CDATA[<session lang="ca" version="2.0"><library closed="false"><mtext style="color:#ffc800" xml:lang="ca">variables</mtext><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k1</mi><mo>=</mo><mi>aleatori</mi><mfenced><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mn>1</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>3</mn></mtd></mtr></mtable></mfenced></mfenced></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><apply><csymbol definitionURL="http://www.wiris.com/XML/csymbol">if</csymbol><mrow><mi>k1</mi><mo>=</mo><mn>1</mn></mrow><mtable><mtr><mtd><apply><csymbol definitionURL="http://www.wiris.com/XML/csymbol">repeat</csymbol><mtable><mtr><mtd><mi>a11</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>2</mn><mo>,</mo><mn>19</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mo>(</mo><mo>{</mo><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>}</mo><mo>)</mo></mtd></mtr><mtr><mtd><mi>a12</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>19</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mo>(</mo><mo>{</mo><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>}</mo><mo>)</mo></mtd></mtr><mtr><mtd><mi>a1</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>3</mn><mo>,</mo><mn>19</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mo>(</mo><mo>{</mo><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>}</mo><mo>)</mo></mtd></mtr><mtr><mtd><mi>a21</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>19</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mo>(</mo><mo>{</mo><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>}</mo><mo>)</mo></mtd></mtr><mtr><mtd><mi>a22</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>19</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mo>(</mo><mo>{</mo><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>}</mo><mo>)</mo></mtd></mtr><mtr><mtd><mi>a2</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>19</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mo>(</mo><mo>{</mo><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>}</mo><mo>)</mo></mtd></mtr><mtr><mtd><mi>A1</mi><mo>=</mo><mfenced><mtable><mtr><mtd><mi>a11</mi></mtd><mtd><mi>a12</mi></mtd></mtr><mtr><mtd><mi>a21</mi></mtd><mtd><mi>a22</mi></mtd></mtr></mtable></mfenced></mtd></mtr><mtr><mtd><mi>A2</mi><mo>=</mo><mfenced><mtable><mtr><mtd><mi>a11</mi></mtd><mtd><mi>a12</mi></mtd><mtd><mi>a1</mi></mtd></mtr><mtr><mtd><mi>a21</mi></mtd><mtd><mi>a22</mi></mtd><mtd><mi>a2</mi></mtd></mtr></mtable></mfenced></mtd></mtr></mtable><mrow><mfenced><mrow><mi>rang</mi><mo>(</mo><mi>A1</mi><mo>)</mo><mo>=</mo><mn>2</mn></mrow></mfenced><mo>∧</mo><mo>(</mo><mi>a11</mi><mo><</mo><mi>a21</mi><mo>)</mo></mrow></apply></mtd></mtr><mtr><mtd><mi>s1</mi><mo>=</mo><mfrac><mfenced close="|" open="|"><mtable><mtr><mtd><mi>a1</mi></mtd><mtd><mi>a12</mi></mtd></mtr><mtr><mtd><mi>a2</mi></mtd><mtd><mi>a22</mi></mtd></mtr></mtable></mfenced><mfenced close="|" open="|"><mtable><mtr><mtd><mi>a11</mi></mtd><mtd><mi>a12</mi></mtd></mtr><mtr><mtd><mi>a21</mi></mtd><mtd><mi>a22</mi></mtd></mtr></mtable></mfenced></mfrac></mtd></mtr><mtr><mtd><mi>s2</mi><mo>=</mo><mfrac><mfenced close="|" open="|"><mtable><mtr><mtd><mi>a11</mi></mtd><mtd><mi>a1</mi></mtd></mtr><mtr><mtd><mi>a21</mi></mtd><mtd><mi>a2</mi></mtd></mtr></mtable></mfenced><mfenced close="|" open="|"><mtable><mtr><mtd><mi>a11</mi></mtd><mtd><mi>a12</mi></mtd></mtr><mtr><mtd><mi>a21</mi></mtd><mtd><mi>a22</mi></mtd></mtr></mtable></mfenced></mfrac></mtd></mtr><mtr><mtd><mi>sol</mi><mo>=</mo><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>s1</mi><mo>,</mo><mi>s2</mi></mtd></mtr></mtable></mfenced></mtd></mtr></mtable><apply><csymbol definitionURL="http://www.wiris.com/XML/csymbol">if</csymbol><mrow><mi>k1</mi><mo>=</mo><mn>2</mn></mrow><mtable><mtr><mtd><mi>a11</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>2</mn><mo>,</mo><mn>9</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mo>(</mo><mo>{</mo><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>}</mo><mo>)</mo></mtd></mtr><mtr><mtd><mi>a12</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>9</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mo>(</mo><mo>{</mo><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>}</mo><mo>)</mo></mtd></mtr><mtr><mtd><mi>a1</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>9</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mo>(</mo><mo>{</mo><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>}</mo><mo>)</mo></mtd></mtr><mtr><mtd><mi>m</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>2</mn><mo>,</mo><mn>6</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mo>(</mo><mo>{</mo><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>}</mo><mo>)</mo></mtd></mtr><mtr><mtd><mi>a21</mi><mo>=</mo><mi>m</mi><mo>*</mo><mi>a11</mi></mtd></mtr><mtr><mtd><mi>a22</mi><mo>=</mo><mi>m</mi><mo>*</mo><mi>a12</mi></mtd></mtr><mtr><mtd><mi>a2</mi><mo>=</mo><mi>m</mi><mo>*</mo><mi>a1</mi></mtd></mtr><mtr><mtd><mi>sol</mi><mo>=</mo><mn>1</mn></mtd></mtr></mtable><mtable><mtr><mtd><mi>a11</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>9</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mo>(</mo><mo>{</mo><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>}</mo><mo>)</mo></mtd></mtr><mtr><mtd><mi>a12</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>9</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mo>(</mo><mo>{</mo><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>}</mo><mo>)</mo></mtd></mtr><mtr><mtd><mi>a1</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>9</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mo>(</mo><mo>{</mo><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>}</mo><mo>)</mo></mtd></mtr><mtr><mtd><mi>n</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>2</mn><mo>,</mo><mn>6</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mo>(</mo><mo>{</mo><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>}</mo><mo>)</mo></mtd></mtr><mtr><mtd><mi>a21</mi><mo>=</mo><mi>n</mi><mo>*</mo><mi>a11</mi></mtd></mtr><mtr><mtd><mi>a22</mi><mo>=</mo><mi>n</mi><mo>*</mo><mi>a12</mi></mtd></mtr><mtr><mtd><mi>a2</mi><mo>=</mo><mo>(</mo><mi>n</mi><mo>+</mo><mn>2</mn><mo>)</mo><mo>*</mo><mi>a1</mi></mtd></mtr><mtr><mtd><mi>sol</mi><mo>=</mo><mn>2</mn></mtd></mtr><mtr><mtd/></mtr></mtable></apply></apply></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>e1</mi><mo>=</mo><mi>a11</mi><mo>*</mo><mi>x</mi><mo>+</mo><mi>a12</mi><mo>*</mo><mi>y</mi><mo>=</mo><mi>a1</mi></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>e2</mi><mo>=</mo><mi>a21</mi><mo>*</mo><mi>x</mi><mo>+</mo><mi>a22</mi><mo>*</mo><mi>y</mi><mo>=</mo><mi>a2</mi></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>w1</mi><mo>=</mo><mi>mcm</mi><mo>(</mo><mi>a11</mi><mo>,</mo><mi>a21</mi><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f1</mi><mo>=</mo><mfrac><mi>w1</mi><mfenced close="|" open="|"><mi>a11</mi></mfenced></mfrac></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f2</mi><mo>=</mo><mfrac><mi>w1</mi><mfenced close="|" open="|"><mi>a21</mi></mfenced></mfrac></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>e21</mi><mo>=</mo><mi>f1</mi><mo>*</mo><mi>e1</mi></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>e22</mi><mo>=</mo><mi>f2</mi><mo>*</mo><mi>e2</mi></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b1</mi><mo>=</mo><mi>f1</mi><mo>*</mo><mi>a11</mi></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b2</mi><mo>=</mo><mi>f2</mi><mo>*</mo><mi>a21</mi></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><apply><csymbol definitionURL="http://www.wiris.com/XML/csymbol">if</csymbol><mrow><mi>a11</mi><mo>*</mo><mi>a21</mi><mo>></mo><mn>0</mn></mrow><mtable><mtr><mtd><mi>z1</mi><mo>=</mo><mo>"</mo><mi>igual</mi><mo>"</mo></mtd></mtr><mtr><mtd><mi>z2</mi><mo>=</mo><mo>"</mo><mi>restem</mi><mo> </mo><mi>la</mi><mo> </mo><mn>2</mn><mi>a</mi><mo> </mo><mi>de</mi><mo> </mo><mi>la</mi><mo> </mo><mn>1</mn><mi>a</mi><mo>"</mo></mtd></mtr></mtable><mtable><mtr><mtd><mi>z1</mi><mo>=</mo><mo>"</mo><mi>diferent</mi><mo>"</mo></mtd></mtr><mtr><mtd><mi>z2</mi><mo>=</mo><mo>"</mo><mi>les</mi><mo> </mo><mi>sumem</mi><mo>"</mo></mtd></mtr></mtable></apply></math></input></command></group></library><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>e1</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mi>x</mi><mo>+</mo><mn>5</mn><mo>*</mo><mi>y</mi><mo>=</mo><mn>1</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>e2</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo>*</mo><mi>x</mi><mo>-</mo><mn>10</mn><mo>*</mo><mi>y</mi><mo>=</mo><mn>0</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>w1</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mi>f1</mi><mo>,</mo><mi>f2</mi><mo>)</mo></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo>,</mo><mn>1</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>e21</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>2</mn><mo>*</mo><mi>x</mi><mo>+</mo><mn>10</mn><mo>*</mo><mi>y</mi><mo>=</mo><mn>2</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>e22</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo>*</mo><mi>x</mi><mo>-</mo><mn>10</mn><mo>*</mo><mi>y</mi><mo>=</mo><mn>0</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn></math></output></command></group><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command></group></session>]]></wirisCasSession><correctAnswers><correctAnswer>#sol</correctAnswer></correctAnswers><assertions><assertion name="syntax_expression"/><assertion name="equivalent_symbolic"/></assertions><options><option name="tolerance">10^(-4)</option><option name="relative_tolerance">false</option><option name="precision">4</option><option name="implicit_times_operator">false</option><option name="times_operator">·</option><option name="imaginary_unit">i</option></options><localData><data name="inputField">popupEditor</data><data name="gradeCompound">and</data><data name="gradeCompoundDistribution"></data><data name="casSession"/></localData></question> Per eliminar x, el mcm de #a11 i #a21 és #w1.

Així doncs, multipliquem

la 1a per «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo mathvariant=¨bold¨ mathcolor=¨#007F00¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#007F00¨»f«/mi»«mn mathvariant=¨bold¨ mathcolor=¨#007F00¨»1«/mn»«mo mathvariant=¨bold¨ mathcolor=¨#007F00¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#007F00¨»§#8594;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#007F00¨»§#8201;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#007F00¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#007F00¨»e«/mi»«mn mathvariant=¨bold¨ mathcolor=¨#007F00¨»21«/mn»«/math»
la 2a per «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo mathvariant=¨bold¨ mathcolor=¨#007F00¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#007F00¨»f«/mi»«mn mathvariant=¨bold¨ mathcolor=¨#007F00¨»2«/mn»«mo mathvariant=¨bold¨ mathcolor=¨#007F00¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#007F00¨»§#8594;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#007F00¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#007F00¨»e«/mi»«mn mathvariant=¨bold¨ mathcolor=¨#007F00¨»22«/mn»«/math»

i com que #b1 i #b2 tenen signe #z1, #z2 per eliminar x.

]]> Si trobem solució, el sistema és compatible determinat.

Si surt 0x+0y=0, el sistema és compatible indeterminat.

Si surt 0x+0y=No-zero el sistema és incompatible.

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