8.06 Solving problems with 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:

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

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.

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.

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}xy	Write the formula
\displaystyle 33	\displaystyle =	\displaystyle \dfrac{1}{2}\times 6y	Substitute the values for A and x
\displaystyle 33	\displaystyle =	\displaystyle 3y	Evaluate \dfrac{1}{2}\times 6
\displaystyle \dfrac{33}{3}	\displaystyle =	\displaystyle \dfrac{3y}{3}	Divide both sides by 3
\displaystyle 11	\displaystyle =	\displaystyle y	Evaluate
\displaystyle y	\displaystyle =	\displaystyle 11 \text{ m}	Swap sides

The length of the other diagonal is 11 m.

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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.

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