A flow diagram has an input value x and an output value y. The rule is to multiply by 9 and then add 8. If the input is 3, what is the output?

Here are the solutions to the flow diagram questions:

Step 1: Complete the flow diagram by calculating missing input and output values.
The rule for the flow diagram is "$\times 9$" then "$+ 8$". This can be written as $y = (x \times 9) + 8$.

• For input $x=3$: $$y = (3 \times 9) + 8 = 27 + 8 = \boxed{\textbf{35}}$$
• For input $x=4$: $$y = (4 \times 9) + 8 = 36 + 8 = \boxed{\textbf{44}}$$
• For input $x=6$: $$y = (6 \times 9) + 8 = 54 + 8 = \boxed{\textbf{62}}$$
• For input $x=7$: $$y = (7 \times 9) + 8 = 63 + 8 = \boxed{\textbf{71}}$$
• For output $y=89$: $$(x \times 9) + 8 = 89$$ $$x \times 9 = 89 - 8$$ $$x \times 9 = 81$$ $$x = 81 \div 9 = \boxed{\textbf{9}}$$
• For output $y=17$: $$(x \times 9) + 8 = 17$$ $$x \times 9 = 17 - 8$$ $$x \times 9 = 9$$ $$x = 9 \div 9 = \boxed{\textbf{1}}$$

ii. What are the input values?
The input values are the numbers that go into the rule.
The input values are $\boxed{\textbf{1, 3, 4, 6, 7, 9}}$.

iii. What are the output values?
The output values are the results after applying the rule to the inputs.
The output values are $\boxed{\textbf{17, 35, 44, 62, 71, 89}}$.

iv. What is the rule?
The rule is explicitly stated in the flow diagram.
The rule is $\boxed{\textbf{multiply by 9, then add 8}}$.

vi. What will the output values be if the rule is "$+ 2 \times 7$"?
We will use the input values from the completed flow diagram: $1, 3, 4, 6, 7, 9$.
The rule "$+ 2 \times 7$" means add the input to the product of 2 and 7. So, $y = x + (2 \times 7) = x + 14$.

• For input $x=1$: $$y = 1 + (2 \times 7) = 1 + 14 = \boxed{\textbf{15}}$$
• For input $x=3$: $$y = 3 + (2 \times 7) = 3 + 14 = \boxed{\textbf{17}}$$
• For input $x=4$: $$y = 4 + (2 \times 7) = 4 + 14 = \boxed{\textbf{18}}$$
• For input $x=6$: $$y = 6 + (2 \times 7) = 6 + 14 = \boxed{\textbf{20}}$$
• For input $x=7$: $$y = 7 + (2 \times 7) = 7 + 14 = \boxed{\textbf{21}}$$
• For input $x=9$: $$y = 9 + (2 \times 7) = 9 + 14 = \boxed{\textbf{23}}$$

The output values would be $\boxed{\textbf{15, 17, 18, 20, 21, 23}}$.