Equations

Equations and graphs

Consider this table of values of $x$ and $y$ .

$x$	$y$
0	20
1	30
2	40
3	50

Notice that for every increase of $1$ in the $x$ the $y$ increases by $10$ . This constant difference shows that there is a linear relationship between $x$ and $y$ . If we plot the values on the axes we get the straight line in the graph above.

Gradient

The gradient of the line is $10$ . The gradient of any line can be calculated using the formula rise over run (either by reading off the graph or using any two coordinates from the table).

$gradient = m =$ $\frac{rise}{run}$ $=$ $\frac{y_2 - y_1}{ x_2 - x_1}$

e.g. Point 1: $(x_1,y_1) =$ $(1,30)$ and Point 2: $(x_1,y_1) =$ $(3,50)$

$gradient = m =$ $\frac{50 - 30}{3 - 1}$ $=$ $\frac{20}{ 2}$ $= 10$

Straight line equations

The general form of a straight line is $y = mx + c$

Where $m$ is the gradient of the line and $c$ is the $y$ intercept (i.e. where the line crosses the $y$ axis).
This form is handy for sketching a line and for reading off the gradient.

e.g. Sketch the line $y = -\frac{1}{2}x + 10$

The $gradient =$ $-\frac{1}{2}$ and the $y$ $intercept = 10$

Note that sometimes we need to rearrange an equation into the form $y = mx + c$ .

e.g. Find the gradient and y intercept of the line $2y - 6x = 4$

$2y = 4 + 6x$

$y = 2 + 3x$

The line will cut the $y$ axis at $2$ with a gradient of $3$ .

Use the desmos online calculator to draw different straight lines to understand the relationship between the equation and the graph.

Special graphs

Horizontal lines

Line: $y= 7$

Gradient = 0

Vertical Lines

Line: $x=6$

Gradient = undefined

Parallel Lines

Lines: $y=x+4$ and $y=x+1$

The gradient of parallel lines are equal.

Tables and graphs drag and drop activity

Complete the following interactive activity

Further information

Press the Printer Friendly button at the top left-hand corner to download a printable handout
Khan academy uses video to explain another worked example of drawing a line with slope and intercept and a set of practice problems that you can use to review your understanding.

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