5.01 Model systems of equations

Ray wants to know how much his pet cat weighs. He could just place the cat down on a scale, but if you own a cat you probably know that's unlikely to happen. He can however stand on the scales while holding the cat. This way, he will get the combined weight of himself plus the cat.

Suppose Ray found that the combined mass was 92 kg. At this stage he still doesn't actually know how much the cat weighs, he could weigh 90 kg and the cat weighs 2 kg (possibly), he could weigh 50 and the cat weighs 42 kg (highly unlikely), or any other number of combinations. He needs at least one more measure of mass before he can work it out.

What Ray can do is weigh himself and then subtract his mass from the combined mass, giving the mass of the cat. If he found his own mass to be 88 kg, how much does the cat weigh?

We can represent this information graphically as follows:

By taking away Ray's mass, 88 kg, from the combined mass of 92 kg, we get 92 - 88 = 4, which means the cat must have a mass of 4 kg.

Graphically, we can represent this as follows:

In this chapter, we will look at solving questions like this with up to two different unknown objects, in which case we will need two unique 'formulas'.

Examples

Example 1

Scales 1 and 2 are in perfect balance. How many As are needed to balance scale 3?

Worked Solution

Create a strategy

Use the information from scales 1 and 2. Find the number of As each B and C is equal to.

Apply the idea

\displaystyle A	\displaystyle =	\displaystyle B	From scale 1.
\displaystyle 3B	\displaystyle =	\displaystyle 5C	From scale 2.
\displaystyle 5C	\displaystyle =	\displaystyle 3A	Substitute B with A.

So there are 3 As needed to balance 5 Cs in scale 3.

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Example 2

Each item has a price. The total price of each row of items is shown alongside it.

We want to work out the price of each item.

a

What is the price of one football?

Worked Solution

Create a strategy

Use the information from the first row of the picture to find the price of one ball.

Apply the idea

On the first row, we only have one football, so the one footbal is equal to \$15.

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b

What is the price of one jersey?

Worked Solution

Create a strategy

Write and solve the equation that represents the second row.

Apply the idea

In the second row, we have 3 jerseys and 1 football. The total price for the second row is \$66 and 1 \text{ football} = \$15.

Let J be price of one jersey and F as the price of one football.

\displaystyle 3J + F	\displaystyle =	\displaystyle 66	Write the equation
\displaystyle 3J + 15	\displaystyle =	\displaystyle 66	Substitute the value of F
\displaystyle 3J	\displaystyle =	\displaystyle 51	Subtract both sides by 15
\displaystyle J	\displaystyle =	\displaystyle \$17	Divide both sides by 3

The price of one jersey is \$17.

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c

What is the price of one pair of boots?

Worked Solution

Create a strategy

Write and solve the equation that represents the third row.

Apply the idea

In the third row, we have 1 pair of boots, 1 jersey, and 1 football. The total price for the third row is \$52, 1 \text{ football} = \$15, and 1 \text{ jersey} = \$17.

Let B represents the price of one pair of boots, J be the price of one jersey, and F as the price of one football.

\displaystyle B + J + F	\displaystyle =	\displaystyle 52	Write the equation
\displaystyle B + 17 + 15	\displaystyle =	\displaystyle 52	Substitute the values of J \text{ and } F
\displaystyle B + 32	\displaystyle =	\displaystyle 52	Add 17 \text{ and } 15
\displaystyle B	\displaystyle =	\displaystyle \$20	Subtract both sides by 32

The price of one pair of boots is \$20.

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Example 3

The combined masses of some animals are shown in the table below. What is the combined mass of a pig, a rabbit, and a cat?

Worked Solution

Create a strategy

Write an equation for each row, add all three rows together and solve.

Apply the idea

Let C represents the mass of a cat, R for the mass of a rabbit, and P as the mass of a pig.

\displaystyle C + R	\displaystyle =	\displaystyle 12	Write the equation for the first row
\displaystyle C + P	\displaystyle =	\displaystyle 17	Write the equation for the second row
\displaystyle R + P	\displaystyle =	\displaystyle 25	Write the equation for the third row
\displaystyle (C + R) + (C + P) + (R + P)	\displaystyle =	\displaystyle 12 + 17 + 25	Add the three equations
\displaystyle 2C + 2R + 2P	\displaystyle =	\displaystyle 54	Simplify
\displaystyle 2 (C + R + P)	\displaystyle =	\displaystyle 54	Factor out 2 on the left-side of the equation
\displaystyle C + R + P	\displaystyle =	\displaystyle 27 \text{ kg}	Divide both sides by 2

The combined mass of a pig, a rabbit, and a cat is 27 \text{ kg}.

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