Atividade de matemática sobre escalamento de sistemas lineares
1º Calcule m e n tal que as equações matriciais representem sistemas lineares equivalentes.
( 2 5 ) ( x ) ( 9 ) ( m 5 ) ( x ) ( 0 )
( 3 -1 ) . ( y ) = ( 4 ) e (4 -n) . ( y ) = ( 5 )
Soluções para a tarefa
Respondido por
40
Bom dia!
Solução!
![\begin{cases}
4x+10y=18\\
15x-10y=20
\end{cases}\\\\\\\\\
4x+15x=18+20\\\\\
19x=38\\\\\
x= \dfrac{38}{19} \\\\\\\
\boxed{x=2}\\\\\\\\
2x+5y=9\\\\
2.2+5y=9\\\\\
4+5y=9\\\\\
5y=9-4\\\\\
5y=5\\\\\
y= \dfrac{5}{5}\\\\\
\boxed{y=1}](https://tex.z-dn.net/?f=%5Cbegin%7Bcases%7D%0A4x%2B10y%3D18%5C%5C%0A15x-10y%3D20%0A%5Cend%7Bcases%7D%5C%5C%5C%5C%5C%5C%5C%5C%5C%0A4x%2B15x%3D18%2B20%5C%5C%5C%5C%5C%0A19x%3D38%5C%5C%5C%5C%5C%0Ax%3D+%5Cdfrac%7B38%7D%7B19%7D+%5C%5C%5C%5C%5C%5C%5C%0A%5Cboxed%7Bx%3D2%7D%5C%5C%5C%5C%5C%5C%5C%5C%0A2x%2B5y%3D9%5C%5C%5C%5C%0A2.2%2B5y%3D9%5C%5C%5C%5C%5C%0A4%2B5y%3D9%5C%5C%5C%5C%5C%0A5y%3D9-4%5C%5C%5C%5C%5C%0A5y%3D5%5C%5C%5C%5C%5C%0Ay%3D+%5Cdfrac%7B5%7D%7B5%7D%5C%5C%5C%5C%5C%0A%5Cboxed%7By%3D1%7D+ "\begin{cases}
4x+10y=18\\
15x-10y=20
\end{cases}\\\\\\\\\
4x+15x=18+20\\\\\
19x=38\\\\\
x= \dfrac{38}{19} \\\\\\\
\boxed{x=2}\\\\\\\\
2x+5y=9\\\\
2.2+5y=9\\\\\
4+5y=9\\\\\
5y=9-4\\\\\
5y=5\\\\\
y= \dfrac{5}{5}\\\\\
\boxed{y=1}")
Vamos substituir o valor de x e y no segundo sistema para determinarmos o valor de m e n.
![\begin{pmatrix} m & 5 \\ 4 & -n \\ \end{pmatrix}.\begin{pmatrix} x \\ y \\ \end{pmatrix}=\begin{pmatrix} 0 \\ 5 \\ \end{pmatrix}\\\\\\
Sistema~~equivalente\\\\\
\begin{cases}
mx+5y=0\\
4x-ny=5
\end{cases}\\\\\\\\
Substituindo!\\\\\\\\
x=2~~~~~~y=1\\\\\
\begin{cases}
m(2)+5(1)=0\\
4(2)-n(1)=5
\end{cases}\\\\\\\\
2m+5=0\\\\\
2m=-5\\\\\
\boxed{m= -\dfrac{5}{2}}\\\\\\
8+n=5\\\\\
n=5-8\\\\
\boxed{n=-3}](https://tex.z-dn.net/?f=%5Cbegin%7Bpmatrix%7D+m+%26amp%3B+5+%5C%5C+4+%26amp%3B+-n+%5C%5C+%5Cend%7Bpmatrix%7D.%5Cbegin%7Bpmatrix%7D+x+%5C%5C+y+%5C%5C+%5Cend%7Bpmatrix%7D%3D%5Cbegin%7Bpmatrix%7D+0+%5C%5C+5+%5C%5C+%5Cend%7Bpmatrix%7D%5C%5C%5C%5C%5C%5C%0A%0ASistema%7E%7Eequivalente%5C%5C%5C%5C%5C%0A%0A%5Cbegin%7Bcases%7D%0Amx%2B5y%3D0%5C%5C%0A4x-ny%3D5%0A%5Cend%7Bcases%7D%5C%5C%5C%5C%5C%5C%5C%5C%0ASubstituindo%21%5C%5C%5C%5C%5C%5C%5C%5C%0Ax%3D2%7E%7E%7E%7E%7E%7Ey%3D1%5C%5C%5C%5C%5C%0A%0A%5Cbegin%7Bcases%7D%0Am%282%29%2B5%281%29%3D0%5C%5C%0A4%282%29-n%281%29%3D5%0A%5Cend%7Bcases%7D%5C%5C%5C%5C%5C%5C%5C%5C%0A2m%2B5%3D0%5C%5C%5C%5C%5C%0A2m%3D-5%5C%5C%5C%5C%5C%0A%5Cboxed%7Bm%3D+-%5Cdfrac%7B5%7D%7B2%7D%7D%5C%5C%5C%5C%5C%5C%0A8%2Bn%3D5%5C%5C%5C%5C%5C%0An%3D5-8%5C%5C%5C%5C%0A%5Cboxed%7Bn%3D-3%7D "\begin{pmatrix} m & 5 \\ 4 & -n \\ \end{pmatrix}.\begin{pmatrix} x \\ y \\ \end{pmatrix}=\begin{pmatrix} 0 \\ 5 \\ \end{pmatrix}\\\\\\
Sistema~~equivalente\\\\\
\begin{cases}
mx+5y=0\\
4x-ny=5
\end{cases}\\\\\\\\
Substituindo!\\\\\\\\
x=2~~~~~~y=1\\\\\
\begin{cases}
m(2)+5(1)=0\\
4(2)-n(1)=5
\end{cases}\\\\\\\\
2m+5=0\\\\\
2m=-5\\\\\
\boxed{m= -\dfrac{5}{2}}\\\\\\
8+n=5\\\\\
n=5-8\\\\
\boxed{n=-3}")
Para que os sistemas sejam equivalentes!
Bom dia!
Bons estudos!
Solução!
Vamos substituir o valor de x e y no segundo sistema para determinarmos o valor de m e n.
Para que os sistemas sejam equivalentes!
Bom dia!
Bons estudos!
Perguntas interessantes