Question #196687

Change the following equations in to augmented matrix

x-y+2z=1

3x-y+5z=-2

4x+2y+(x2-8)z=(x+2)


And determine values of x where:-

There is no solution

And where there is exactly one solution

And where there infinitely many solutions


1
Expert's answer
2021-05-24T12:29:30-0400

Augmented matrix 


A=[1121315242x28x+2]A=\begin{bmatrix} 1 & -1 & 2 & 1 \\ 3 & -1 & 5 & -2 \\ 4 & 2 & x^2-8 & x+2 \end{bmatrix}



R2=R23R1R_2=R_2-3R_1

[1121021542x28x+2]\begin{bmatrix} 1 & -1 & 2 & 1 \\ 0 & 2 & -1 & -5 \\ 4 & 2 & x^2-8 & x+2 \end{bmatrix}



R3=R34R1R_3=R_3-4R_1

[1121021506x216x2]\begin{bmatrix} 1 & -1 & 2 & 1 \\ 0 & 2 & -1 & -5 \\ 0 & 6 & x^2-16 & x-2 \end{bmatrix}



R2=R2/2R_2=R_2/2

[1121011/25/206x216x2]\begin{bmatrix} 1 & -1 & 2 & 1 \\ 0 & 1 & -1/2 & -5/2 \\ 0 &6 & x^2-16 & x-2 \end{bmatrix}



R1=R1+R2R_1=R_1+R_2

[103/23/2011/25/206x216x2]\begin{bmatrix} 1 & 0 & 3/2 & -3/2 \\ 0 & 1 & -1/2 & -5/2 \\ 0 & 6 & x^2-16 & x-2 \end{bmatrix}



R3=R36R2R_3=R_3-6R_2

[103/23/2011/25/200x213x+13]\begin{bmatrix} 1 & 0 & 3/2 & -3/2 \\ 0 & 1 & -1/2 & -5/2 \\ 0 & 0 & x^2-13 & x+13 \end{bmatrix}

I) no solution


x213=0x+130=>x=±13\begin{matrix} x^2-13=0 \\ x+13\not=0 \end{matrix}=>x=\pm\sqrt{13}



Ii) exactly one solution


x2130=>x±13x^2-13\not=0=>x\not=\pm\sqrt{13}



Iii) infinitely many solutions


x213=0x+13=0=>x\begin{matrix} x^2-13=0 \\ x+13=0 \end{matrix}=>x\in \varnothing




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