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# Filling in a table of values

## 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.

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