Developing linear equations
There are many problems that can be solved by identifying two variables that have a linear relationship to one another. When a linear relation is identified, a linear equation can be formed to solve the given problem.
Finding the rule from a table of values
Tables are used everywhere in mathematics, usually to show data for two or more related quantities (represented by variables).
When given a table of values relating two quantities, it is often useful to figure out if there is a relationship between them and what the relationship is. If we can find a relationship, it can be used to predict future values and patterns.
This section teaches how to develop a linear equation from a given table of values, where the two quantities have a linear relationship.
Worked example
Consider the four pictures below, side by side. Develop a linear equation that relates the number of petals visible in each picture ($y$y) with the corresponding number of flowers ($x$x).
Looking at the flowers above we can use a table to more easily understand the pattern that relates the number of flowers to the number of petals:
| Flowers ($x$x) | $1$1 | $2$2 | $3$3 | $4$4 |
| Petals ($y$y) | $5$5 | $10$10 | $15$15 | $20$20 |
In this pattern, $x$x represents the step we are at (that is, the number of flowers) and $y$y represents the total number of petals at that step.
Notice that $y$y is increasing by $5$5 each time - in particular, the value of $y$y is always equal to $5$5 times the value of $x$x. We can express this as the algebraic rule $y=5x$y=5x.
Now that we have this rule, we can use it to predict future results. For example, if we wanted to know the total number of petals when there were $10$10 flowers, we have that $x=10$x=10 and so $y=5\times10=50$y=5×10=50 petals.
Practice question
Question 1
Use the table of values below to write an equation for $g$g in terms of $f$f.
| $f$f | $4$4 | $5$5 | $6$6 | $7$7 | $8$8 |
|---|---|---|---|---|---|
| $g$g | $8$8 | $10$10 | $12$12 | $14$14 | $16$16 |
Finding the rule from a worded description
When constructing a linear equation from a worded sentence, look for key terms such as "sum", "minus", or "is equal to". Most importantly, identify what question is being asked that requires a solution.
Practice questions
Question 2
The product of $5$5 and the sum of $x$x and $7$7 equals $50$50.
Construct an equation and find the value of $x$x.
Question 3
To manufacture sofas, the manufacturer has a fixed cost of $\$27600$$27600 plus a variable cost of $\$170$$170 per sofa. Find $n$n, the number of sofas that need to be produced so that the average cost per sofa is $\$290$$290.