Simultaneous equations：練習問題

Find the sum of the \(x\)-coordinates of the points where the curve \(y = x^2\) intersects the line \(y = 4\).

Solve the following simultaneous equations:
\(y - x + 3 = 0\)
\(x^2 - 3xy + y^2 + 19 = 0\)

Find the sum of the \(y\)-coordinates of the two points of intersection.

The line \(y = mx + 2\) is a tangent to the curve \(y = x^2 + 6\). Find the possible values of the constant \(m\).

Find the point of intersection of the two lines represented by the equations \(y = 4x - 1\) and \(y = x + 5\).

The line \(y = x + 1\) intersects the curve \(y = x^2 - x - 2\) at two points. Calculate the sum of the \(y\)-coordinates of these two points.

Find the coordinates of the points of intersection where the line \(y = 4 - x\) meets the curve \(y = \frac{3}{x}\).

Solve the following simultaneous equations for \(x\) and \(y\):
\(x + 2y = 7\)
\(x^2 - 4y^2 = 21\)

The area of a rectangle is \(48 \text{ cm}^2\) and its perimeter is \(28 \text{ cm}\). By forming and solving a pair of simultaneous equations, find the dimensions of the rectangle.

Solve the following simultaneous equations:
\(x - 2y = 1\)
\(xy = 15\)

Solve the simultaneous equations, giving your answers in their simplest form:
\(y - 2x = 3\)
\(2x^2 - 3xy + y^2 = 1\)