Linear Equations From Table of Values

Hi everyone. In this post, I cover the math topic of determining linear equations from a table of values. As long a table of values meets some conditions we can determine a linear function that passes these points.

Pixabay Image Source

Topics

---

• What Is A Linear Function?
• Determining Linear Functions From A Table Of Values

What Is A Linear Function?

---

A linear function is a function where the inputs correspond in such a way where the rate of increase or decrease stays the same every time the input value increases by one. The general form of the linear function is:

where

• x is the independent variable or input
• m is the slope or the rate of increase/decrease for every 1 unit increase in x
• y is the output from x given a slope m

Example One

One apple costs 50 cents. Two apples would cost 100 cents. For every 1 apple increase the cost for the apples goes up by 50 cents. Let n be the number of apples. The cost for the apples (A) would be A = 0.5 x n.

Example Two

Consider a second example. A taxi fare has a cost for going into the cab and a variable cost that depends on the number of kilometres traveled. I am using price data from numbeo.com for Toronto. Getting in a taxi costs $4.25 CAD and it costs $2.00 CAD for every kilometre traveled.

Let y be the cost of taxi trip and let x be the number of kilometres travelled. The linear function for this Toronto taxi fare case would be:

The 4.25 CAD amount is called the y-intercept. It is the y-value when x = 0. In this case, $4.25 CAD would be the entry cost or base fee of the taxi ride. Each kilometre travelled costs $2 each on top of the $4.25 initial cost.

Pixabay Image Source

Determining Linear Functions From A Table Of Values

---

What if all you have are a table of values? From the table of values, you would need to determine the slope and the y-intercept to obtain a corresponding linear function.

Here are a few examples. These two examples are not too difficult (I think).

Example One

Given a table of values, what is the equation of the line that passes these points?

x	y
-2	-11
-1	-6
0	-1
1	4
2	9

For every 1 unit increase for x, the value of y increases by 5. The rate of change or slope is 5. This five is the slope or the value for m.

With the equation of a line, we have:

The y-intercept can be quickly determined from the table of values. It is the value for y when x = 0. Negative one would be the y-intercept.

For this example, the equation of the line is y = 5x -1. A graph of this line would look something like this Desmos screenshot.

Example Two

Determine the equation of the line that passes these 3 points given in the table below.

x	y
1	40
2	30
3	20

From the table, a one unit increase for x corresponds to a decrease of 10 for y. The slope here for the equation of a line is just negative ten.

So far we have:

The value of the y-intercept b needs to be determined. Select an (x, y) point from table of values, substitute accordingly for x and y and then solve for b. I choose (1, 40) as my (x, y) point.

With the value of b being 50 and a slope of -10, the equation of the line that passes through the 3 points from the table is y = -10x + 50.

Thank you for reading.