Answer to Question #196275 in Linear Algebra for anuj

Question #196275

Use Cramer’s rule to solve for y without solving for x, z and w in the system 2w + x + y + z = 3 −8w − 7x − 3y + 5z = −3 w + 4x + y + z = 6 w + 3x + 7y − z = 1 


1
Expert's answer
2021-05-21T15:47:40-0400
2w+x+y+z=38w7x3y+5z=3w+4x+y+z=6w+3x+7yz=1\begin{matrix} 2w + x + y + z = 3 \\ −8w − 7x − 3y + 5z = −3 \\ w + 4x + y + z = 6 \\ w + 3x + 7y − z = 1 \end{matrix}

Write the matrix for the system and check that the determinant is not equal zero.


A=[2111873514111371]A=\begin{bmatrix} 2 & 1 & 1 & 1 \\ -8 & -7 & -3 & 5 \\ 1 & 4 & 1 & 1 \\ 1 & 3 & 7 & -1 \\ \end{bmatrix}

detA=2111873514111371\det A=\begin{vmatrix} 2 & 1 & 1 & 1 \\ -8 & -7 & -3 & 5 \\ 1 & 4 & 1 & 1 \\ 1 & 3 & 7 & -1 \\ \end{vmatrix}

=2735411371835111171=2\begin{vmatrix} -7 & -3 & 5 \\ 4 & 1 & 1 \\ 3 & 7 & -1 \\ \end{vmatrix}-\begin{vmatrix} -8 & -3 & 5 \\ 1 & 1 & 1 \\ 1 & 7 & -1 \\ \end{vmatrix}

+875141131873141137+\begin{vmatrix} -8 & -7 & 5 \\ 1 & 4 & 1 \\ 1 & 3 & -1 \\ \end{vmatrix}-\begin{vmatrix} -8 & -7 & -3 \\ 1 & 4 & 1 \\ 1 & 3 & 7 \\ \end{vmatrix}


735411371=43571+7531\begin{vmatrix} -7 & -3 & 5 \\ 4 & 1 & 1 \\ 3 & 7 & -1 \\ \end{vmatrix}=-4\begin{vmatrix} -3 & 5 \\ 7 & -1 \\ \end{vmatrix}+\begin{vmatrix} -7 & 5 \\ 3 & -1 \\ \end{vmatrix}

7337=4(32)+(8)(40)=160-\begin{vmatrix} -7 & -3 \\ 3 &7 \\ \end{vmatrix}=-4(-32)+(-8)-(-40)=160


835111171=3571+8511\begin{vmatrix} -8 & -3 & 5 \\ 1 & 1 & 1 \\ 1 & 7 & -1 \\ \end{vmatrix}=-\begin{vmatrix} -3 & 5 \\ 7 & -1 \\ \end{vmatrix}+\begin{vmatrix} -8 & 5 \\ 1 & -1 \\ \end{vmatrix}

8317=(32)+(3)(53)=88-\begin{vmatrix} -8 & -3 \\ 1 &7 \\ \end{vmatrix}=-(-32)+(3)-(-53)=88




875141131=841317531\begin{vmatrix} -8 & -7 & 5 \\ 1 & 4 & 1 \\ 1 & 3 & -1 \\ \end{vmatrix}=-8\begin{vmatrix} 4 & 1 \\ 3 & -1 \\ \end{vmatrix}-\begin{vmatrix} -7 & 5 \\ 3 & -1\\ \end{vmatrix}

+7541=8(7)(8)+(27)=37+\begin{vmatrix} -7 & 5 \\ 4 &1 \\ \end{vmatrix}=-8(-7)-(-8)+(-27)=37




873141137=841377337\begin{vmatrix} -8 & -7 & -3 \\ 1 & 4 & 1 \\ 1 & 3 & 7 \\ \end{vmatrix}=-8\begin{vmatrix} 4 & 1 \\ 3 & 7 \\ \end{vmatrix}-\begin{vmatrix} -7 & -3 \\ 3 & 7 \\ \end{vmatrix}

+7341=8(25)(40)+(5)=155+\begin{vmatrix} -7 & -3\\ 4 & 1 \\ \end{vmatrix}=-8(25)-(-40)+(5)=-155detA=2(160)88+37(155)\det A=2(160)-88+37-(-155)

detA=4240\det A=424\not=0



B=[2131873514611311]B=\begin{bmatrix} 2 & 1 & 3 & 1 \\ -8 & -7 & -3 & 5 \\ 1 & 4 & 6 & 1 \\ 1 & 3 & 1 & -1 \\ \end{bmatrix}

detB=2131873514611311\det B=\begin{vmatrix} 2 & 1 & 3 & 1 \\ -8 & -7 & -3 & 5 \\ 1 & 4 & 6 & 1 \\ 1 & 3 & 1 & -1 \\ \end{vmatrix}

=2735461311835161111=2\begin{vmatrix} -7 & -3 & 5 \\ 4 & 6 & 1 \\ 3 & 1 & -1 \\ \end{vmatrix}-\begin{vmatrix} -8 & -3 & 5 \\ 1 & 6 & 1 \\ 1 & 1 & -1 \\ \end{vmatrix}

+3875141131873146131+3\begin{vmatrix} -8 & -7 & 5 \\ 1 & 4 & 1 \\ 1 & 3 & -1 \\ \end{vmatrix}-\begin{vmatrix} -8 & -7 & -3 \\ 1 & 4 & 6 \\ 1 & 3 & 1 \\ \end{vmatrix}




735461311=43511+67531\begin{vmatrix} -7 & -3 & 5 \\ 4 & 6 & 1 \\ 3 & 1 & -1 \\ \end{vmatrix}=-4\begin{vmatrix} -3 & 5 \\ 1 & -1 \\ \end{vmatrix}+6\begin{vmatrix} -7 & 5 \\ 3 & -1 \\ \end{vmatrix}

7331=4(2)+6(8)(2)=42-\begin{vmatrix} -7 & -3 \\ 3 &1 \\ \end{vmatrix}=-4(-2)+6(-8)-(2)=-42




835161111=35618511\begin{vmatrix} -8 & -3 & 5 \\ 1 & 6 & 1 \\ 1 & 1 & -1 \\ \end{vmatrix}=\begin{vmatrix} -3 & 5 \\ 6 & 1 \\ \end{vmatrix}-\begin{vmatrix} -8 & 5 \\ 1 & 1 \\ \end{vmatrix}

8316=33(13)(45)=25-\begin{vmatrix} -8 & -3 \\ 1 & 6 \\ \end{vmatrix}=-33-(-13)-(-45)=25




875141131=841317531\begin{vmatrix} -8 & -7 & 5 \\ 1 & 4 & 1 \\ 1 & 3 & -1 \\ \end{vmatrix}=-8\begin{vmatrix} 4 & 1 \\ 3 & -1 \\ \end{vmatrix}-\begin{vmatrix} -7 & 5 \\ 3 & -1\\ \end{vmatrix}

+7541=8(7)(8)+(27)=37+\begin{vmatrix} -7 & 5 \\ 4 &1 \\ \end{vmatrix}=-8(-7)-(-8)+(-27)=37




873146131=846317331\begin{vmatrix} -8 & -7 & -3 \\ 1 & 4 & 6 \\ 1 & 3 & 1 \\ \end{vmatrix}=-8\begin{vmatrix} 4 & 6 \\ 3 & 1 \\ \end{vmatrix}-\begin{vmatrix} -7 & -3 \\ 3 & 1 \\ \end{vmatrix}+7346=8(14)(2)(30)=80+\begin{vmatrix} -7 & -3\\ 4 & 6 \\ \end{vmatrix}=-8(-14)-(2)-(30)=80




detB=2(42)25+3(37)80=78\det B=2(-42)-25+3(37)-80=-78

y=detBdetA=78424=39212y=\dfrac{\det B}{\det A}=\dfrac{-78}{424}=-\dfrac{39}{212}



y=39212y=-\dfrac{39}{212}



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