Which of the following describe or determine a single straight line?

\(4x - 2y = 6\)

\(y = 2\)

The points \((1,2)\) and \((0,-1)\)

\(y = \frac{3}{2}x\)

The points \((-1, -4)\), \((3, 7)\), and \((8, 8)\)

\(y = 3 - 2x\)

The point \((0,-1)\) and the constant gradient \(3\)

\(x=-2\)

\(y = x^2 + 2\)

\(x = 7y + 5\)

\(y - 8 = 3(x - 3)\)

The points \(\left(\frac{1}{2},2\right)\), \((1,1)\), and \(\left(\frac{3}{2}, 0\right)\)

\(y^2 = x^2\)

The point \((3,3)\) and the direction specified by the vector \(\big( \begin{smallmatrix}1\\2\end{smallmatrix}\big)\)

\(\frac{1}{3}y - x + \frac{1}{3} = 0\)

\(xy = 1\)

\(y^2 - 4xy + 4x^2 = 0\)

How did you decide?

You might find it helpful to try sketching some of these. You could use graphing software such as Desmos.

Which of the descriptions in question 1 represent the same straight line?

What ways of describing straight lines have been used in question 1? Can you give any other ways to describe a unique straight line?

Now consider the equation \(y=4x+1\), which describes a straight line. Use some of the ways of describing straight lines you have identified to give alternative descriptions of this straight line.

Why might one description be more useful than another? Can you identify any advantages or disadvantages of any of these representations?