## Part I. How Linear Equations relate to Tables Of Values

### Equations as Relationships

The equation of a line expresses a relationship between x and y values on the coordinate plane. For instance, the equation y = x expresses a relationship where every x value has the exact same y value. The equation y = 2x expresses a relationship in which every y value is double the x value, and y = x +1 expresses a relationship in which every y value is 1 greater than the x value.

*So what about a Table Of Values?*

Since, as we just wrote, every equation is a relationship of x and y values, we can create a table of values for any line, these are just the x and y values that are true for the given line. In other words, a table of values is simply some of the points that are on the line.

Let's See Some Examples

##### Example 1

**Equation: ** y = x + 1

**Table of Values**

X Value | Equation | Y value |

y = x +1 | ||

3 | y = (3) +1 | 4 |

4 | y = (4) +1 | 5 |

5 | y = (5) +1 | 6 |

6 | y = (6) + 1 | 7 |

##### Example 2

**Equation: ** y = 3x + 2

**Table of Values**

X Value | Equation | Y value |

y = 3x + 2 | ||

1 | y = 3(1) + 2 | 5 |

2 | y = 3(2) + 2 | 7 |

3 | y = 3(3) + 2 | 11 |

4 | y = 3(4) + 2 | 14 |

So, to create a table of values for a line, just pick a set of x values, substitute them into the equation and evaluate to get the y values.

**Practice** Creating a Table of Values

Create a table of values of the equation y = 5x + 2

Create the table and choose a set of x values

X Value | Equation | Y value |

y = 5x +2 | ||

1 | ||

2 | ||

3 | ||

4 |

Substitute each x value (left side column) into the equation

X Value | Equation | Y value |

y = 5x +2 | ||

1 | y = 5(1) +2 | |

2 | y = 5(2) +2 | |

3 | y = 5(3) +2 | |

4 | y = 5(4) +2 |

Evaluate the equation (middle column) to arrive at the y value.

X Value | Equation | Y value |

1 | y = 5(1) +2 | 7 |

2 | y = 5(2) +2 | 12 |

3 | y = 5(3) +2 | 17 |

4 | y = 5(4) +2 | 22 |

An Optional step, if you want, you can omit the middle column from your table, since the table of values is really just the x and y pairs.

(We used the middle column simply to help us get the y values)

X Value | Y Value |

1 | 7 |

2 | 12 |

3 | 17 |

4 | 22 |

Create a table of values of the equation y = −6x + 2

Create the table and choose a set of x values

X Value | Equation | Y value |

y = −6x + 2 | ||

1 | ||

2 | ||

3 | ||

4 |

Substitute each x value (left side column) into the equation

X Value | Equation | Y value |

y = −6x + 2 | ||

1 | y = −6(1) + 2 | |

2 | y = −6(2) + 2 | |

3 | y = −6(3) + 2 | |

4 | y = −6(4) + 2 |

Evaluate the equation (middle column) to arrive at the y value.

X Value | Equation | Y value |

1 | y = −6x + 2 | -4 |

2 | y = −6x + 2 | -10 |

3 | y = −6x + 2 | -16 |

4 | y =y = −6x + 2 | -22 |

An Optional step, if you want, you can omit the middle column from your table, since the table of values is really just the x and y pairs .(We used the middle column simply to help us get the y values)

X Value | Y value |

1 | -4 |

2 | -10 |

3 | -16 |

4 | -22 |

Create a table of values of the equation y = −6x − 4

Create the table and choose a set of x values

X Value | Equation | Y value |

y = −6x − 4 | ||

1 | ||

2 | ||

3 | ||

4 |

Substitute each x value (left side column) into the equation

X Value | Equation | Y value |

1 | y = −6(1) − 4 | |

2 | y = −6(2) − 4 | |

3 | y = −6(3) − 4 | |

4 | y = −6(4) − 4 |

Evaluate the equation (middle column) to arrive at the y value.

X Value | Equation | Y value |

1 | y = −6(1) − 4 | -10 |

2 | y = −6(2) − 4 | -16 |

3 | y = −6(3) − 4 | -22 |

4 | y = −6(4) − 4 | -28 |

An Optional step, if you want, you can omit the middle column from your table, since the table of values is really just the x and y pairs. (We used the middle column simply to help us get the y values)

X Value | Y value |

1 | -10 |

2 | -16 |

3 | -22 |

4 | -28 |

## Part II. Writing Equation from Table of Values

Often, students are asked to Write the equation of a line from a table of values. To solve this kind of problem, simply chose any 2 points on the table and follow the normal steps for writing the equation of a line from 2 points.

Write the equation of a line from the table of values below.

Choose any two x,y pairs from the table and calculate the slope. Since, I like to work with easy, small numbers I chose (0,3) and (1,7) .

X Value | Y value |

0 |
3 |

1 |
7 |

2 | 11 |

3 | 15 |

Substitute slope into the slope intercept form of a line

y = mx +b

y = 4x +b

Find the value of 'b' in the slope intercept equation

y = mx +b

y = 4x +b

Since our table gave us the point (0,3) we know that 'b' is 3. Remember 'b' is the y-intercept which, luckily, was supplied to us in the table.

**Answer: **y = 4x + 3

If you'd like, you could check your answer by substituting the values from the table into your equation. Each and every x,y pair from the table should work with your answer.

Write the equation from the table of values provided above.

Choose any two x,y pairs from the table and calculate the slope. I chose (2 , 8) and (4,9).

X | Y |

2 |
8 |

4 |
9 |

6 | 10 |

Substitute slope into the slope intercept form of a line

y = mx +b

y = ½x +b

Find the value of 'b' in the slope intercept equation

Now that we know the value of b , we can substitute it into our equation

**Answer: **y = ½x + 7

If you'd like, you could check your answer by substituting the values from the table into your equation. Each and every x,y pair from the table should work with your answer.

Write the equation from the table of values provided above.

Choose any two x,y pairs from the table and calculate the slope. I chose (2 , 8) and (4,9).

X | Y |

3 |
2 |

6 |
0 |

9 | -2 |

Substitute slope into the slope intercept form of a line

y = mx +b

y = x +b

Find the value of 'b' in the slope intercept equation

Now that we know the value of b , we can substitute it into our equation.

**Answer: ** y = x + 4

If you'd like, you could check your answer by substituting the values from the table into your equation. Each and every x,y pair from the table should work with your answer.

**Challenge Problem**

Why can you **not** write the equation of a line from the table of values below?

**Remember:**The slope of a line never changes