Given the flow diagram below: Determine the values of a and b. The rule of a pattern is given as: Output = (Input × 3) – 2. Calculate the output when the input is 5. Write the rule as a number sentence in terms of n. Explain how this rule is similar to: Output = 3n – 2

The input for $a$ is 4.

$$a = (4 \times 2) + 3$$ $$a = 8 + 3$$ $$a = 11$$

The output for $b$ is 15.

$$(b \times 2) + 3 = 15$$

Subtract 3 from both sides:

$$b \times 2 = 15 - 3$$ $$b \times 2 = 12$$

Divide by 2:

$$b = \frac{12}{2}$$ $$b = 6$$ $$\boxed{a = 11, b = 6}$$

8.2 The rule of a pattern is given as: $\text{Output} = (\text{Input} \times 3) - 2$

8.2.1 Calculate the output when the input is 5.

$$\text{Output} = (5 \times 3) - 2$$ $$\text{Output} = 15 - 2$$ $$\text{Output} = 13$$ $$\boxed{13}$$

8.2.2 Write the rule as a number sentence in terms of $n$.

$$\text{Output} = (n \times 3) - 2$$

This can be written more concisely as:

$$\text{Output} = 3n - 2$$ $$\boxed{\text{Output} = 3n - 2}$$

8.2.3 Explain how this rule is similar to: $\text{Output} = 3n - 2$

The rule $\text{Output} = (\text{Input} \times 3) - 2$ is similar to $\text{Output} = 3n - 2$ because in algebra, the multiplication symbol ($\times$) between a number and a variable (like $3 \times n$) is often omitted and written as $3n$. Also, the variable $n$ is used to represent the "Input".

$$\boxed{\text{The rules are identical. "Input" is replaced by the variable } n, \text{ and } 3 \times n \text{ is written as } 3n.}$$