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# 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.

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.

### Horizontal lines

Line: $y= 7$

### Vertical Lines

Line: $x=6$

### 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.