Building and Solving

Equations

Slideshow 23, MathematicsMr Richard SasakiRoom 307

Objectives• Practice building simple equations

based on worded examples• Understand how one unknown can

be used for two circumstances that relate

Building EquationsWe built a few simple equations in Chapter 2. Let’s try building some more.ExampleA man pays for chocolate bars with Yen and receives Yen change. How much did each chocolate bar cost? State any assumptions.300=2 𝑥+40260=2 𝑥𝑥=130

Therefore, a chocolate bar costs Yen.

What did we assume?Both chocolate bars have the same value.

Building EquationsLet’s try another example.ExampleA boy goes to a shop and has a choice. He can buy either three bottles of milk and have Yen change or buy two bottles of milk with Yen left over. How much does a bottle of milk cost?3 𝑥+120=2𝑥+210𝑥=90

Therefore, a bottle of milk costs Yen.

Here we assumed…

Both bottles cost the same amount.

AnswersThe cans of cola are the same price.

Both drinks cost the same amount.

𝑥+18,000=3𝑥+2000⟹𝑅𝑖𝑐𝑒𝑐𝑜𝑜𝑘𝑒𝑟=8,000円The rice cookers are the same price. Both prizes have the same value.

Each option has the same value and the chocolates and bottles of wine cost the same amount.

Two Unknowns?How do we deal with two variables / unknowns that directly effect one another?

ExampleA girl spends exactly 200 Yen and buys an apple and an orange. The apple costs Yen. State the amount an orange costs in terms of Yen.

200𝑌𝑒𝑛𝑥𝑌𝑒𝑛(200−𝑥 )𝑌𝑒𝑛Apple Orange

The orange costs Yen.

Two Unknowns?Let’s try another.ExampleA cat and dog eat food worth 3,000 kcal (calories) between them. If the cat consumes kcal, state the amount the dog eats.So…

Cat FoodDog Food⇒𝑥 +¿ ¿3000(3000−𝑥 )

This means that the dog eats worth of food.If the cat ate , what would the dog eat?

3000−800=2200𝑘𝑐𝑎𝑙

Two Unknowns?Let’s try a harder question.ExampleAt a shop, a man buys some bananas at and oranges at each. He buys a combination of of them and pays . How many bananas and oranges did he buy?Consider the number of bananas to be .Therefore, the total cost of the bananas is .40𝑥So, the total cost of the oranges is .60 (12−𝑥 )Total cost:40𝑥+60 (12−𝑥 )=540⇒ 40 𝑥+720−60 𝑥=540⇒−20 𝑥=−180(bananas) bananas, oranges

Answers - Easy(170−𝑥)𝑚𝑒𝑡𝑟𝑒𝑠

170−75=95𝑚𝑒𝑡𝑟𝑒𝑠100 𝑥+180 (7− 𝑥 )=940⇒𝑥=4 juice and milk

5 𝑥+8 (5−𝑥 )=31⇒𝑥=3 hours walking and hours jogging

45𝑥+70 (10−𝑥 )=525⇒𝑥=7 computers running and printers

Answers - Hard5 𝑥+7 (840−𝑥 )=5100⇒𝑥=390Coffee: Yen, Cake: Yen

6 𝑥−9=4 𝑥+9⇒𝑥=9 children, sweets

3 𝑥−10=2𝑥+20⇒𝑥=30 boys, counters

3 𝑥−6=2𝑥+4⇒𝑥=10 guests, cups of tea