Solve: `a x+b y=c ,\ \ \ \ b x+a y=1+c`

Solve the following system of equations by method of cross-multiplication: x+a y=b ,\ \ \ \ a x-b y=c

Solve the following system of equations by method of cross-multiplication: b x+c y=a+b ,\ \ \ \ and a x(1/(a-b)-1/(a+b))+c y(1/(b-a)-1/(b+a))=(2a)/(a+b) .

The lines a x+b y=c, b x+c y=a and c x+a y=b are concurrent, if

If x+y+z=0 prove that |a x b y c z c y a z b x b z c x a y|=x y z|a b cc a bb c a|

Solve: a^2x+b^2y=c^2; b^2x+a^2y=d^2

Prove that: |a b a x+b y b c b x+c y a x+b y b x+c y0|=(b^2-a c)(a x^2+2b x y+c y^2) .

Prove that |[a x-b y-c z, a y+b x, c x+a z], [a y+b x, b y-c z-a x, b z+c y],[c x+a z, b z+c y, c z-a x-b y]|=(x^2+y^2+z^2)(a^2+b^2+c^2)(a x+b y+c z)dot

Prove that |a x-b y-c z a y+b x c x+a z a y+b x b y-c z-a x b z+c y c x+a z b z+c y c z-a x-b y|=(x^2+y^2+z^2)(a^2+b^2+c^2)(a x+b y+c z)dot

Prove that |a x-b y-c z a y+b x c x+a z a y+b x b y-c z-a x b z+c y c x+a z b z+c y c z-a x-b y|=(x^2+y^2+z^2)(a^2+b^2+c^2)(a x+b y+c z)dot

Prove that |[a x-b y-c z, a y+b x, c x+a z], [a y+b x, b y-c z-a x, b z+c y],[c x+a z, b z+c y, c z-a x-b y]|=(x^2+y^2+z^2)(a^2+b^2+c^2)(a x+b y+c z)dot