## 22.3 Filling in a table of values

To compile a table of values for a given equation, we follow these steps:

- Choose an \(x\)-value from the first row in the table.
- Replace the \(x\) in the equation with this value.
- Calculate the corresponding \(y\)-value.
- Fill in this \(y\)-value in the second row of the table.
- Repeat for all the \(x\)-values.

## Worked Example 22.3: Filling in a table of values

Fill in the table of values below for the given equation. Your answers should be exact (no rounding).

\[y = 5x - 3\]\(x\) | \(- 5\) | \(- 2\) | \(- 1\) | \(5\) | \(10\) |

\(y\) |

### Calculate the value of \(y\) for \(x = - 5\).

All the \(y\)-values are missing from the table, so you can use the corresponding \(x\)-values from the table and calculate the \(y\)-values using the equation. Remember that the equation encodes all of the information about the relationship between \(x\) and \(y\).

For the first blank space in the table, substitute \(x = - 5\) into the equation and calculate the missing value.

\[\begin{align} y &= 5x - 3 \\ &= 5( - 5) - 3 \\ &= - 25 - 3 \\ &= - 28 \end{align}\]For \(x = - 5\), the missing \(y\)-value in the table is \(y = - 28\).

### Calculate the value of \(y\) for \(x = - 2\).

\[\begin{align} y &= 5x - 3 \\ &= 5( - 2) - 3 \\ &= - 10 - 3 \\ &= - 13 \end{align}\]For \(x = - 2\), the \(y\)-value is \(y = - 13\).

### Calculate the value of \(y\) for \(x = - 1\).

\[\begin{align} y &= 5x - 3 \\ &= 5x - 3 \\ &= 5( - 1) - 3 \\ &= - 5 - 3 \\ &= - 8 \end{align}\]For \(x = - 1\), the \(y\)-value is \(y = - 8\).

### Calculate the value of \(y\) for \(x = 5\).

\[\begin{align} y &= 5x - 3 \\ &= 5(5) - 3 \\ &= 25 - 3 \\ &= 22 \end{align}\]For \(x = 5\), the \(y\)-value is \(y = 22\).

### Calculate the value of \(y\) for \(x = 10\).

\[\begin{align} y &= 5x - 3 \\ &= 5(10) - 3 \\ &= 50 - 3 \\ &= 47 \end{align}\]For \(x = 10\), the \(y\)-value is \(y = 47\).

### Complete the table.

The completed table looks like this:

\(x\) | \(- 5\) | \(- 2\) | \(-1\) | \(5\) | \(10\) |

\(y\) | \(-28\) | \(-13\) | \(-8\) | \(22\) | \(47\) |

Think about the number of possible \(x\)-values for the given equation, \(y = 5x - 3\).

The table of values above shows five ordered pairs that all solve the equation, but there are many, many more. You can pick any number you want for \(x\), and the equation will tell you the value of \(y\) that belongs with it. In this way, you get another ordered pair that solves the equation. There are an infinite number of \(x\)-values that you can use, so there is an infinite number of solutions to the equation.

Remember that a solution to the equation is an ordered pair that fits into the equation. In other words, a solution is a combination of \(x\)- and \(y\)-values that agree with the equation.