Straight-line graphs：練習問題

Line \(L_1\) has the equation \(y = 3x + 5\). If line \(L_2\) is perpendicular to \(L_1\), which of the following could be the gradient of \(L_2\)?

Find the equation of the line passing through the point \((1, 4)\) that is parallel to the line \(2x + 3y = 6\).

Line \(L_1\) has the equation \(y = 2x - 4\). Line \(L_2\) passes through the point \((0, 6)\) and is perpendicular to \(L_1\). Find the coordinates of the point of intersection of \(L_1\) and \(L_2\).

Find the gradient of the straight line passing through the points \((3, -2)\) and \((-1, 6)\).

Find the equation of the line that passes through the point \((2, -1)\) and is parallel to the line \(4x - 2y = 7\).

The line \( L_1 \) has the equation \( 4x - 2y = 7 \). Find the gradient of a line \( L_2 \) that is perpendicular to \( L_1 \).

The points \( P(k, 3) \) and \( Q(2, -1) \) are such that the length of the line segment \( PQ \) is \( \sqrt{20} \) units. Find the two possible values of the constant \( k \).

When a graph of \( xy \) is plotted against \( x^2 \), a straight line is obtained which passes through the points \( (1, 4) \) and \( (3, 10) \). Express \( y \) in terms of \( x \).

Variables \(x\) and \(y\) satisfy the equation \(y = A e^{kx}\), where \(A\) and \(k\) are constants. When \(\ln y\) is plotted against \(x\), a straight line is obtained that passes through the points \((2, 3.5)\) and \((6, 5.5)\).
(a) Find the value of \(k\) and the value of \(A\), giving your answers correct to 2 decimal places.
(b) Find the value of \(y\) when \(x = 10\).
(c) Find the value of \(x\) when \(y = 100\).