If you’ve completed the previous sub-module on determining whether a relation is linear or not, you’ve seen the three ways in which we can represent linear relations:

1. Numerically - with a table of values

2. Graphically - with a graph

3. Algebraically - with an equation or formula

In this sub-module, we’ll relate these representations so that we are able to move fluidly between all three.

In the last module, we briefly went over how to use a linear equation to fill a table of values.

When we have ordered pairs with x and corresponding y-values, it is important to remember that x is the independent variable and as such, y depends on x.

With that in mind, we can use a linear equation to create a table of values by selecting various x-values and solving for the corresponding y-values, and putting the results in the table.

Check out the following video to see this in action:

If we are given a linear equation in the form y = mx + b, we can create a graph in two ways:

1. By using the equation to construct a table of values, and then plotting each point

2. By using the slope and y-intercept to plot various points directly

In the two videos below, we’ll go over the second method mentioned above.

To use the first method, check out the sections on going from an equation to a table of values, and then going from a table of values to a graph.

As we learned in the last sub-module, we sometimes have to manipulate a linear equation to get it in the y = mx + b form. This process is crucial if you want to use the second method of plotting the graph directly from the equation.

Obtaining a graph from a table of values involves plotting each point (each ordered x-y pair). Check out the video below to see this in action.

Going from a table of values to an equation requires that one finds two pieces of information:

1. The slope (m)

2. The y-intercept (b)

The y-intercept is the y-value when x = 0.

The slope can be found by calculating the first differences of the y-values. In the video below, we see that this involves subtracting the previous y-value from a selected y-value. In more simple terms, you subtract the 1st y-value from the 2nd y-value, the 2nd y-value from the 3rd y-value, and so on.

You can revisit the previous sub-module if you need more practice with first differences.

To obtain a filled table of values using a graph, various points on the graph must be selected and their respective x-y values placed in the table.

You can chose any points you want from the graph, but I like to pull "nice" points from the graph - usually whole numbers or numbers without complicated decimals like 4.5.

In order to get the equation for a linear relation from a graph, you need to find two key pieces of information: the slope (m) and the y-intercept (b).

Finding the y-intercept is quite easy; the y-intercept is the y-value of the point where the line crosses the y-axis or the y-value when x = 0.

To find the slope, we can use the formula for slope where slope is equal to the change in y over the change in x, as seen below.