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2. Linear Relations and Equations
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AustraliaVIC
VCE 11 General 2023

2.04 Develop linear equations

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Introduction

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.

Linear equation 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.

Consider the four pictures below, side by side. Develop a linear equation that relates the number of petals visible in each picture (y) with the corresponding number of flowers (x).

Flowers in groups of 1, 2, 3 and 4. Each flower has 8 petals.

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:

Number of flowers1234
Number of petals5101520

In this pattern, y represents the step we are at (that is, the number of flowers) and x represents the total number of petals at that step.

Notice that y is increasing by 5 each time - in particular, the value of y is always equal to 5 times the value of x. We can express this as the algebraic rule 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 flowers, we have that x=10 and so y=5\times 10=50 petals.

Examples

Example 1

Use the table of values below to write an equation for g in terms of f.

f34567
g68101214
Worked Solution
Create a strategy

Determine the number that must be multiplied to row f to get to the corresponding numbers in row g.

Apply the idea

To determine the number to be multiplied to row f, we divide some values of g by f.

\displaystyle \dfrac{g_1}{f_1}\displaystyle =\displaystyle \dfrac{6}{3}Divide the first value of g by first value of f
\displaystyle =\displaystyle 2Evaluate
\displaystyle \dfrac{g_2}{f_2}\displaystyle =\displaystyle \dfrac{8}{4}Divide the second value of g by second value of f
\displaystyle =\displaystyle 2Evaluate

This means that 2 must be multiplied to row f to get the corresponding numbers in row g.

So, the equation for g in terms of f is given by:g=2 \times f

Idea summary

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.

Linear equation 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.

Examples

Example 2

The product of 5 and the sum of x and 7 equals 50. Construct an equation and find the value of x.

Worked Solution
Create a strategy

Rewrite the sentence as a mathematical equation. Remember that product means multiplication and sum means addition.

Apply the idea

The mathematical equation of the sentence is given by:5\left(x+7\right)=50

Solving for x, we have

\displaystyle 5\times x + 5\times 7\displaystyle =\displaystyle 50Use distributive property of multiplication
\displaystyle 5x + 35\displaystyle =\displaystyle 50Evaluate the multiplication
\displaystyle 5x + 35-35\displaystyle =\displaystyle 50-35Subtract 35 from both sides of equation
\displaystyle 5x\displaystyle =\displaystyle 15Combine like terms
\displaystyle \dfrac{5x}{5}\displaystyle =\displaystyle \dfrac{15}{5}Divide both sides by 5
\displaystyle x\displaystyle =\displaystyle 3Evaluate the division

Example 3

To manufacture sofas, the manufacturer has a fixed cost of \$27\,600 plus a variable cost of \$170 per sofa. Find n, the number of sofas that need to be produced so that the average cost per sofa is \$290.

Worked Solution
Create a strategy

Determine the mathematical equation for the average cost of producing n sofas. Use the fact that the total cost of producing n sofas is equal to the sum of the fixed cost and the variable cost.

Apply the idea

The mathematical equation for the total cost of producing n sofas is given by:27\,600+170n

Since the average cost per sofa is \$290 and the average cost is equal to the total cost divide by the total number of things, n, then solving for n we have:

\displaystyle 290\displaystyle =\displaystyle \frac{27\,600+170n}{n}Write the mathematical equation for average cost
\displaystyle 290n\displaystyle =\displaystyle 27\,600+170nMultiply both sides by n
\displaystyle 290n-170n\displaystyle =\displaystyle 27\,600+170n-170nSubtract 170n from both sides
\displaystyle 120n\displaystyle =\displaystyle 27\,600Combine like terms
\displaystyle \dfrac{120n}{120}\displaystyle =\displaystyle \dfrac{27\,600}{120}Divide both sides by 120
\displaystyle n\displaystyle =\displaystyle 230Evaluate
Idea summary

When constructing a linear equation, it is important to identify what question is being asked that requires a solution. Moreover, look for key terms such as "sum", "minus", or "is equal to" to determine what operations to use to solve the linear equation.

Outcomes

U1.AoS4.3

the forms, rules, graphical images and tables for linear relations and equations

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