5.4 Using equations and tables
Input and output values with equations
Mathematics is like a language, and you need to be able to translate the verbal description of the rule into a mathematical equation, inequality or formula.
For example, the rule is “\(y\) is equal to \(x\) squared”. The question tells us a rule that connects the \(x\) to the \(y\). To start the solution, we must “translate” the words into a maths expression.
The most important thing is to work out which operation the rule tells us to use. In this case, the keyword is squared, which means an exponent of two. The phrase “\(x\) squared” means \(x^{2}\).
The rule translates to this equation: \(y = x^{2}\).
Now, if \(x = 2\), then \(y = (2)^{2} = 4\). We put \(x = 2\) into the equation \(y = x^{2}\) and calculate the answer for \(y\).
Worked Example 5.7: Using algebraic notation to describe rules
Two values are related by the following rule:
“The output value is the sum of the input value and \(1\)”.
What is the output value if the input value is \(−7\)?
Rewrite the words as an equation.
The rule says: “the output value is the sum of the input value and \(1\)”. We need to work out which operation the rule tells us to use. In this case, the keyword is sum, which means addition. So the “sum of the input value and \(1\)” means \(x + 1\).
The rule translates to this equation:
\[y = x + 1\]Place the input value in the equation and find the answer.
With the equation, the only thing left to do is put the input value, \(−7\), in place of \(x\).
\[\begin{align} y &= x + 1 \\ y &= ( - 7) + 1 \\ y &= - 6 \end{align}\]According to the rule, if the input value is \(−7\), the correct output value is \(−6\).
Tables and equations
Remember that a table of values lists two groups of numbers: the input values at the top and the output values at the bottom. The numbers that are above and below each other are related to each other by the equation, where \(x\) is always the input value for an equation, and \(y\) is always the output value. Each pair of \(x\)- and \(y\)-values that correspond to each other and make the equation true is called a solution of the given equation.
- solution (to an equation)
- an ordered pair that fits into the equation
Therefore, a solution is a combination of \(x\)- and \(y\)-values that makes one side of the equation equal to the other side.
Worked Example 5.8: Using equations to find missing values in a table
The table below summarises the equation, \(y = 3x - 1\). However, there is a missing number in the \(y\)-values.
| \(x\) | \(1\) | \(5\) | \(7\) | \(9\) |
| \(y\) | ? | \(14\) | \(20\) | \(26\) |
What is the missing number?
Find the correct input number.
This question asks us to find one of the \(y\)-values for the equation, \(y = 3x - 1\). In order to do this calculation, we must find the correct input value to use. From the table, we have the input value to use: \(x = 1\).
Find the corresponding output value.
To find the corresponding output value, put \(x = 1\) into the equation. Then calculate the answer for \(y\).
\[y = 3x - 1\]Plug in \(x = 1\):
\[\begin{align} y &= 3(1) - 1 \\ &= 3 - 1 \\ &= 2 \end{align}\]The missing value from the table is the output value of \(y = 2\).
Worked Example 5.9: Using equations to find missing values in a table
Fill in the table of values below for the equation, \(y = - x^{2} - 1\).
How many solutions are there for the equation?
| \(x\) | \(−3\) | \(−2\) | \(−1\) | \(0\) | \(1\) | \(2\) |
| \(y\) | \(−10\) | \(−5\) | \(−1\) | \(−5\) |
Find the first input number that has a missing output number.
For the first blank space in the table, put \(x = -1\) into the equation and calculate the missing value:
\[\begin{align} y &= - x^{2} - 1 \\ y &= - ( - 1)^{2} - 1 \\ y &= - 1 - 1 \\ & = - 2 \end{align}\]The minus sign is not part of the square in \(x^{2}\), so when substituting the value for \(x\), keep the minus in front.
The first missing value from the table is \(y = -2\).
Find the next input number with a missing output number.
Repeat the process, but now with \(x = 1\).
\[\begin{align} y &= - x^{2} - 1 \\ y &= - (1)^{2} - 1 \\ y &= - 1 - 1 \\ &= - 2 \end{align}\]The second missing value from the table is \(y = -2\).
Determine the number of solutions.
There is an infinite number of solutions to the equation! The table of values above shows \(6\) ordered pairs that make the equation true. But there are many, many more. You can pick any number you want for \(x\) and the equation will tell you the value of \(y\) which belongs with it to get another ordered pair that solves the equation. If you keep changing the input value, you put into the equation you will get more and more (and more and more …) solutions.
The number of inputs is infinite, so the number of solutions is also infinite, which we can write as \(\infty\).