Solve the simultaneous equations

\(\frac{x}{2y}-\frac{4y}{x}=-1,\qquad x=1-6y.\)

(a) Solve the equation \(x^{\frac13}-x^{\frac16}=2\).

(b) Solve the simultaneous equations \(\lg(x+2y)=0\) and \(x^2+4xy+y=1\).

(a) Given that \(256^{x+y} \times 16^{-2x}=8^{-x+3y}\), show that \(y=3x\).

(b) Hence find the exact solutions of the following simultaneous equations.

\(256^{x+y} \times 16^{-2x}=8^{-x+3y}\)

\(x^2+3y^2=56\)

Solve the following simultaneous equations. \(\begin{aligned} \frac{y}{x} & =\frac{3}{2} \\ \frac{y^{4}}{x^{5}} & =\frac{27}{16} \end{aligned}\)

Solve the following simultaneous equations, giving your answers in the form \(a+b\sqrt7\), where \(a\) and \(b\) are integers.

\(x+3y=11\)

\(x-\sqrt7y=7\)

Solve the following simultaneous equations.

\(x+5y=-4\)

\(3y-xy=6\)

The curve

\(\frac4{x^2}+\frac5{4y^2}=1\)

and the line \(x+2y=0\) intersect at two points. Find the exact distance between these points.

Solve the following simultaneous equations.

\(4x^2+3xy+y^2=8\)

\(xy+4=0\)

Solve the simultaneous equations

\(xy+x^2=15,\qquad y+3x=11.\)

Solve the simultaneous equations. Give your answers in the form \(a+b\sqrt3\), where \(a\) and \(b\) are rational.

\(x+y=3,\qquad 2x-\sqrt3\,y=5.\)

Solve the simultaneous equations

\(x^2+3xy=4,\qquad 2x+5y=4.\)

Do not use a calculator in this question.

Solve the simultaneous equations, giving your answers for both \(x\) and \(y\) in the form \(a+b\sqrt2\), where \(a\) and \(b\) are integers.

\(2x+y=5,\)

\(3x-\sqrt2y=7.\)

Solve the equations

\(y-x=4, \qquad x^2+y^2-8x-4y-16=0.\)

Solve the equations

\(y-x=4, \qquad x^2+y^2-8x-4y-16=0.\)

Solve the simultaneous equations for \(x\) and \(y\), giving each answer in its simplest surd form.

\(\sqrt3x+y=4\)

\(x-2y=5\sqrt3\)