C2.3 Solve equations that involve multiple terms, integers, and decimal numbers in various contexts, and verify solutions.
Activity 1: Solving Equations Using Various Strategies
Ellie received 4 identical bills from her parents and added them to the $10 she already had. Sam received 5 identical bills to those of Ellie. Sam spent $10. Everyone now has the same amount of money. Determine the dollar value of the bills they received.
STRATEGY 1
Representation by Flowchart
Step 1: I represent 1 bill by the letter b
[b] + [b] + [b] + [b] + [10] = [b] + [b] + [b] + [b] + [b] - [10]
I understand that I can remove the same number of [b]s from each side of the equation without changing the equation.
\(\begin{align}[b] + [b] + [b] + [b] - [b] - [b] - [b] + [b] + [10] &= [b] + [b] + [b] + [b] + [b] - [b] - [b] - [b] - [b] - [10] \\ [10] &= [b] - [10]\end{align}\)I have to find a number that I can subtract 10 from to get 10.
I have to do the reverse operation to find this number, that is 10 + 10, which is equal to 20.
\(\begin{align} [b] \rightarrow &[-10] \rightarrow [10] \\ [20] \leftarrow &[+10] \leftarrow [10] \end{align}\)
The unit value of the bills they received is $20. In other words, Ellie received 4 $20 bills and Sam received 5 $20 bills.
Step 2: I check my solution.
\(\begin{align} 4 \times 20 + 10 &= 5 \times 20 \ - 10 \\ 80 + 10 &= 100 \ - 10 \\ 90 &= 90 \end{align}\)
STRATEGY 2
Balance Model
Step 1: I represent the problem with an equation.
4b + 10 = 5b - 10
I understand that if I add or subtract the same amount from each side of the equation, the equation is changed, but the equality remains true. So, I remove 4b from each side.
\(\begin{align}4b + 10 - 4b &= 5b -10 - 4b \\ 10 &= b - 10 \end{align}\)I now have b - 10 on one side of the equation. To keep only b, I need to add +10 to cancel out -10. I have to do the same on the other side to keep the equality.
\(\begin{align} 10 &= b -10 \\ 10 + 10 &= b - 10 + 10 \\ 20 &= b \end{align}\)Step 2: I check my solution.
\(\begin{align} 4(20) + 10 &= 5(20) - 10 \\ 80 + 10 &= 100 - 10 \\ 90 &= 90 \end{align}\)
Activity 2: Solving Equations
Solve the following equations:
- \(3x - 6 + x = -4.3 + 8.5 - 1.1x\)
- \(-2.7z + 3.2 + (3z - 2.1z) = 6.1 - 0.2z + 0.3\)
Activity 3: Water Volume
Three times the volume of a container plus four litres is equivalent to five times the volume minus 6 litres. What is the volume of this container?
Activity 4: Mass
In a certain game, there are large and small tokens. The mass of four small tokens is equal to the mass of one large token.
If the mass of two large tokens plus 15.3 g is equal to the mass of five large tokens plus 6 g, what is the mass of one small token?