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8.06 Solving problems with equations

Lesson

Introduction

Equations are useful because they can help us find values that we don't know yet. This can range from unknown numbers in a number sentence to missing dimensions in a shape. However, in order to use equations to help us with these problems, we will first need to be able to convert our problems into equations and then interpret our results.

Convert problems into equations

Converting our real world problems into mathematical equations that we can solve is very similar to how we convert between word statements and number sentences.

When converting our problems into mathematical language, the most important thing to understand is what the problem is and which operations we can use to represent it.

Sometimes the equations that we make cannot be solved immediately because we do not have enough information. This happens when we end up with more than one unknown value. In equations, this is represented by there being more than one pronumeral.

However, these equations are still useful because they can show us the relationship between the two unknown values.

Consider the following scenario.

Hana and Curt eat a box of 12 biscuits together.

We do not know how many biscuits each person gets but we can still represent this scenario using an equation.

We can represent the number of biscuits Hana gets with the pronumeral h, and we can do the same for Curt with the pronumeral c. Since their total will be equal to 12, we can express this scenario with the equation:h+c=12We can now use this equation to answer some questions we might have.

If Curt gets 5 biscuits, how many does Hana get?

In other words, if c=5, what is the value for h?

We can find the answer by substituting our new value into the equation. Replacing c with 5 in our equation gives us:h+5=12Solving this equation tells us that, when c=5,\,h=7. In other words, if Curt gets 5 biscuits then Hana gets 7.

What if c is equal to some other values? We can represent this using a table of values:

c678910
h65432

As we can see from the table, as the value for c increases, the value for h decreases. In other words, the more biscuits Curt gets, the less Hana gets.

We can see from this example that equations with more than one unknown value can be useful, either by substituting new information into the equation or by testing a range of values to find a relationship between the unknown values.

Examples

Example 1

Consider the word statement 'y is equal to the product of 3 and the sum of x and 4'.

a

Write an equation in the form y=⬚ that describes the word statement.

Worked Solution
Create a strategy

Convert the word statement into a mathematical equation.

Apply the idea

The phrase 'y is equal to' corresponds to the y= part of the equation.

The phrase 'the product of 3 and the sum of x and 4' means we want to multiply 3 with the sum of x and 4.

The phrase 'the sum of x and 4' can be written as x+4.

When we put this together we get 'the product of 3 and x+4' which can be written as 3(x+4).

So we have the equation:y=3(x+4)

b

Using the equation found in part (a), complete the table.

x123510
y
Worked Solution
Create a strategy

Substitute each value of x from the table into the equation.

Apply the idea

Substitute x=1:

\displaystyle y\displaystyle =\displaystyle 3(x+4)Wrtie the equation
\displaystyle y\displaystyle =\displaystyle 3\times(1+4)Substitute x=1
\displaystyle =\displaystyle 3\times5Evaluate the addition
\displaystyle =\displaystyle 15Evaluate

Similarly, if we substitute the other values of x, ( x=2,\, x=3,\, x=5,\, x=10 ), into y=3(x+4), we get:

x123510
y1518212742
Idea summary

Equations with more than one unknown value can be useful, either by substituting values for one unknown into the equation or by testing a range of values to find a relationship between the unknown values.

Formulas

An important application of equations and substitution is for using formulas.

A formula is an equation that defines the relationship between two or more variables, usually representing a real world or mathematical relationship.

In mathematics, we often use formulas to find unknown values based on relationships that we know are true for certain situations.

We know that the area of a rectangle will always be equal to the product of its length and width.

This image shows a rectangle with the length and width labelled.

We can represent this relationship using the equation:A=l \times wwhere A= Area, l= length and w= width.

Using this equation, we can find any of the three variables as long as we know the other two.

Suppose the length is 4 units and the width is 7 units. Substituting this into the equation gives us:A=4\times7Solving this tells us that a rectangle with these dimensions will have an area of 28 square units.

What about a rectangle that has an area of 20 square units and a length of 4 units?

We can find the height of such a rectangle by substituting our known values into the equation giving us:20=4wSolving this tells us that a rectangle with this area and length must have a width of 5 units.

Examples

Example 2

The area of a rhombus is given by the formula A=\dfrac{1}{2}xy, where x and y are the lengths of the diagonals.

A particular rhombus has a short diagonal length x=6 m and area 33 \text{ m}^2.

An image of a rhombus with the diagonals x and y.

Set up an equation and rearrange to solve for the unknown y.

Worked Solution
Create a strategy

Use the formula of the area of a rhombus and substitute the area and x values to solve for y.

Apply the idea
\displaystyle A\displaystyle =\displaystyle \dfrac{1}{2}xyWrite the formula
\displaystyle 33\displaystyle =\displaystyle \dfrac{1}{2}\times 6ySubstitute the values for A and x
\displaystyle 33\displaystyle =\displaystyle 3yEvaluate \dfrac{1}{2}\times 6
\displaystyle \dfrac{33}{3}\displaystyle =\displaystyle \dfrac{3y}{3}Divide both sides by 3
\displaystyle 11\displaystyle =\displaystyle yEvaluate
\displaystyle y\displaystyle =\displaystyle 11 \text{ m}Swap sides

The length of the other diagonal is 11 m.

Example 3

The speed of a plane can be calculated using the formula S=\dfrac{D}{T}, where D is distance travelled, T is time taken and S is speed.

If a plane travels 3600 kilometres in 6 hours, what is its speed?

Worked Solution
Create a strategy

Use the formula for the speed of a plane.

Apply the idea
\displaystyle S\displaystyle =\displaystyle \dfrac{D}{T}Write the formula
\displaystyle =\displaystyle \dfrac{3600}{6}Substitute the values for D and T
\displaystyle =\displaystyle 600\text{ km/hr}Evaluate

The speed of the plane is 600 km/hr.

Idea summary

A formula is an equation that defines the relationship between two or more variables, usually representing a real world or mathematical relationship.

We can use formulas to solve problems by substituting values for some unknowns and solving for the remaining variable.

Outcomes

MA4-10NA

uses algebraic techniques to solve simple linear and quadratic equations

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