Maths reference: Linear relations

What each new figure adds goes in front of $n$; what is there at the start goes on its own. Figure 4 has $3 \times 4 + 1 = 13$ sticks.

A rule turns each input $x$ into an output $y$. Put the input in for $x$: $x = 3$ gives $y = 2 \times 3 + 1 = 7$.

Choose inputs, work out each output from the rule ($y = 2x + 1$ here), and write them in pairs.

Input first, then output. To plot it: across 2 on the $x$-axis, then up 5.

Find the value you know on its axis, go to the line, then read across to the other axis. Sections of a graph increase, decrease or stay constant.

To see what one letter does, change only that letter and keep the others the same. Record the results in a table.

$m$, the number in front of $x$: one step right, the line goes up $m$ (down when $m$ is negative). A bigger $m$ is steeper; $m = 0$ is flat.

$c$, the number on its own: where the line crosses the $y$-axis (at $x = 0$). Changing $c$ moves the line up or down without changing its steepness.

The same $m$ gives the same steepness, so the lines never meet. $y = 2x + 4$ is $y = 2x + 1$ moved up 3.

One letter: every point on $x = 3$ has $x = 3$ (a vertical line); every point on $y = -2$ has $y = -2$ (horizontal).

Draw the boundary $y = 2x + 1$: dashed for $<$ and $>$, solid for $\le$ and $\ge$. Test a point such as $(0, 0)$: true, shade its side; false, the other side.

Draw each side as a line: $y = 2x + 3$ and $y = 11$. Where they meet, both sides are equal, so the $x$ there is the solution.

A starting amount plus a rate: the start is $c$ (4 dollars to get in) and the rate is $m$ (2 dollars per km).

Put both rules on one graph. Where the lines cross, the plans cost the same; on either side, the lower line is cheaper. Check where the model makes sense (no negative gigabytes).