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.

## 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.

## 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`

.

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