5.07 Rewrite expressions to understand relationships
Introduction
In the previous year, we have learned how to represent algebraic expressions. We use algebraic expressions when we want to write a number sentence but don't know all the numbers involved. This time, we will write expressions in different forms to help us solve real world problems and show how the quantities in expressions are related.
Ideas
Rewrite expressions
Simplifying an expression doesn't always make it easier to understand the relationship between two things. Different relationships can be represented by expressions in different but equivalent forms.
Imagine that you have \$10 to spend on after school snacks for a week, and each snack from the vending machine costs \$2.
First, let's consider some different expressions that give the amount of snack money you have left after buying s snacks. 10-2s \quad \quad 2(5-s) \quad \quad 10-s-s \quad \quad -2s+10
Although the expressions appear different, all four expressions represent the amount of money you have left after buying s snacks. Let's interpret the terms of each expression in the context of our situation.
In the expression 10-2s, the number 10 represents the amount of money you have before buying any snacks. The number -2 represents the cost of buying one snack. The term 2s represents the amount of money you have to subtract from your initial amount after buying s snacks.
In the expression 2(5-s), the number 5 represents the total number of snacks you can buy. The expression (5-s) represents the number of snacks you have left, and the 2 represents the amount of money for each snack you have left.
We can use any of the equations, to find the amount of money we have left after buying 4 snacks by substituting this value in the expression.
Using the expression 10-2s we have:
| \displaystyle 10-2s | \displaystyle = | \displaystyle 10-2(4) | Sustitute 4 to the value of x |
| \displaystyle = | \displaystyle 10-8 | Evaluate the product | |
| \displaystyle = | \displaystyle 2 | Evaluate the difference |
We were left with \$2.
If we try to substitute 4 to all the variables in the other expressions, we'll see that we'll get the same answer.
Examples
Example 1
In a particular year, Re-source Waste Recovery produces sewage treatment plants at a cost of \$ u per plant. Re-source Waste Recovery also pays a fee of \$ b per year for its use of the production facilities.
Using only addition, write an expression that represents the total production cost from 1 year if there are 5 sewage treatment plants produced in that year.
Which of the following expressions is also equivalent to the production cost in that year?
Example 2
Xavier travels to a foreign country, where the value of currency is described in terms of the values x and y. When Xavier arrives, he has plenty of coins worth 2y and plenty of notes worth x + y, but none of the several other types of coins or notes.
Xavier goes to a cafe to buy a hot breakfast and wants to work out whether he can buy one without getting any change back.
The display price of the meal is 4x+8y. Rewrite the display price in the form ⬚(x+y)+⬚(2y).
Xavier looks at the rest of menu and thinks about which other items he could buy. Which two of these items could he buy without receiving any change back?
Example 3
Consider the diagram below.
Hermione found the total green shaded area by considering the length and width of the overall rectangle. Complete the values for the expression she found.\text{Area}=⬚(6+⬚)
Hermione found the total green shaded area by adding the areas of the two smaller rectangles. Complete the values for the expression she found.
| \displaystyle \text{Area} | \displaystyle = | \displaystyle 4\times ⬚ + ⬚ \times 4 |
| \displaystyle = | \displaystyle ⬚ + 4x |
Different relationships can be represented by algebraic expressions in different but equivalent forms.