Solve the following system of linear equations in three variables.
x + y + z = 6; 2x + 3y + 4z = 20;
3x + 2y + 5z = 22

Using quadratic formula solve the following equations.
p²x² + (P² -q²) X - q² = 0

Using quadratic formula solve the following equations.9x²-9(a+b)x+(2a²+5ab+2b²)=0

Find the values of k for which the following equation has equal roots.
(k - 12)r + 2(k - 12)x + 2 = 0

Prove that the equation x²(a²+b²)+2x(ac+bd)+(c²+ d²) = 0 has no real root if ad≠bc.

Solve: 5x + 2y =3, 3x + 2y = 5

Find the GCD of \(x^{3}+8 y^{3} \text { and } x+2 y\)

m² - 5m - 14 is an expression. Find out another similar expression such that their HCF is (m - 7) and LCM is m³ - 10m² + 11m + 70

Find the GCD of \(10\left(x^{2}+x-20\right), 15\left(x^{2}-3 x-4\right)\text { and } 20\left(x^{2}+2 x+1\right)\)

Find the LCM of \(a^{2}+3 a+2, a^{2}+5 a+6 \text { and }a^{2}+4 a+4\)

Simplify : \(\frac{3 x^{2}-3 x}{3 x^{3}-6 x^{2}+3 x}\)

Simplify : \(\frac{x^{2}+3 x}{x^{2}-4 x-21}\)

Simplify : \(\frac{2 x^{2}+26 x+84}{2 x^{2}+12 x-14}\)

Simplify : \(\frac{1}{4 x^{2}}+\frac{5}{6 x y^{2}}\)

Simplify : \(\frac{3 x}{x^{2}+3 x-10}-\frac{6}{x^{2}+3 x-10}\)