Sejam v⃗ 1=(1,3), v⃗ 2=(−1,5), v⃗ 3=(4,−3) e v⃗ 4=(3,0). Determine x+y onde v⃗ 1=xv⃗ 2+yv⃗ 4.
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Boa tarde Carol!
Solução!
![\vec{v_{1}}=x(\vec{v_{2}})+y( \vec{v_{4}})\\\\\\\
Substituindo~~os~~vetores~~na~~express\~ao~~fica~~assim!\\\\\\\\
(1,3)=x(-1,5)+y(3,0)\\\\\\\
(1,3)=(-1x,5x)+(3y,0y)\\\\\\\
Veja,saimos~~em~~um~~ sistema~~linear~~com~~duas~~variaveis.\\\\\\
\begin{cases}-1x+3y=1\\\\\
~~5x+0y=3
\end{cases}\\\\\\\\\
5x=3](https://tex.z-dn.net/?f=%5Cvec%7Bv_%7B1%7D%7D%3Dx%28%5Cvec%7Bv_%7B2%7D%7D%29%2By%28+%5Cvec%7Bv_%7B4%7D%7D%29%5C%5C%5C%5C%5C%5C%5C%0ASubstituindo%7E%7Eos%7E%7Evetores%7E%7Ena%7E%7Eexpress%5C%7Eao%7E%7Efica%7E%7Eassim%21%5C%5C%5C%5C%5C%5C%5C%5C%0A%281%2C3%29%3Dx%28-1%2C5%29%2By%283%2C0%29%5C%5C%5C%5C%5C%5C%5C%0A%281%2C3%29%3D%28-1x%2C5x%29%2B%283y%2C0y%29%5C%5C%5C%5C%5C%5C%5C%0AVeja%2Csaimos%7E%7Eem%7E%7Eum%7E%7E+sistema%7E%7Elinear%7E%7Ecom%7E%7Eduas%7E%7Evariaveis.%5C%5C%5C%5C%5C%5C%0A%5Cbegin%7Bcases%7D-1x%2B3y%3D1%5C%5C%5C%5C%5C%0A%7E%7E5x%2B0y%3D3%0A%5Cend%7Bcases%7D%5C%5C%5C%5C%5C%5C%5C%5C%5C%0A5x%3D3%0A%0A%0A "\vec{v_{1}}=x(\vec{v_{2}})+y( \vec{v_{4}})\\\\\\\
Substituindo~~os~~vetores~~na~~express\~ao~~fica~~assim!\\\\\\\\
(1,3)=x(-1,5)+y(3,0)\\\\\\\
(1,3)=(-1x,5x)+(3y,0y)\\\\\\\
Veja,saimos~~em~~um~~ sistema~~linear~~com~~duas~~variaveis.\\\\\\
\begin{cases}-1x+3y=1\\\\\
~~5x+0y=3
\end{cases}\\\\\\\\\
5x=3")
![\boxed{x= \dfrac{3}{5}}\\\\\\\\\
-1(\dfrac{3}{5})+3y=1\\\\\\\\
-\dfrac{3}{5}+3y=1\\\\\\\
-3+15y=5\\\\\\\
15y=8\\\\\\\
\boxed{y= \dfrac{8}{15}}](https://tex.z-dn.net/?f=%5Cboxed%7Bx%3D+%5Cdfrac%7B3%7D%7B5%7D%7D%5C%5C%5C%5C%5C%5C%5C%5C%5C%0A-1%28%5Cdfrac%7B3%7D%7B5%7D%29%2B3y%3D1%5C%5C%5C%5C%5C%5C%5C%5C+%0A-%5Cdfrac%7B3%7D%7B5%7D%2B3y%3D1%5C%5C%5C%5C%5C%5C%5C%0A-3%2B15y%3D5%5C%5C%5C%5C%5C%5C%5C%0A15y%3D8%5C%5C%5C%5C%5C%5C%5C%0A%5Cboxed%7By%3D+%5Cdfrac%7B8%7D%7B15%7D%7D+%0A "\boxed{x= \dfrac{3}{5}}\\\\\\\\\
-1(\dfrac{3}{5})+3y=1\\\\\\\\
-\dfrac{3}{5}+3y=1\\\\\\\
-3+15y=5\\\\\\\
15y=8\\\\\\\
\boxed{y= \dfrac{8}{15}}")
![Logo!\\\\\
~~x~~+~y\\\\\
\dfrac{3}{15}+ \dfrac{8}{15}= \dfrac{11}{15} \\\\\\\
\boxed{Resposta:~~x+y= \dfrac{11}{15}}](https://tex.z-dn.net/?f=Logo%21%5C%5C%5C%5C%5C%0A%7E%7Ex%7E%7E%2B%7Ey%5C%5C%5C%5C%5C%0A+%5Cdfrac%7B3%7D%7B15%7D%2B+%5Cdfrac%7B8%7D%7B15%7D%3D+%5Cdfrac%7B11%7D%7B15%7D+%5C%5C%5C%5C%5C%5C%5C%0A%5Cboxed%7BResposta%3A%7E%7Ex%2By%3D+%5Cdfrac%7B11%7D%7B15%7D%7D "Logo!\\\\\
~~x~~+~y\\\\\
\dfrac{3}{15}+ \dfrac{8}{15}= \dfrac{11}{15} \\\\\\\
\boxed{Resposta:~~x+y= \dfrac{11}{15}}")
Boa tarde!
Bons estudos!
Solução!
Boa tarde!
Bons estudos!
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