Question

Translate the following verbal rules into symbolic rules using \(x\) as the input and \(y\) as the output variable. a. The output is ten more than the input. b. The output is the sum of the input and 12 c. The output is five less than the input. d. The output is six times the input. e. The output is three less than six times the input. f. The output is six times the difference of the input and three. g. The output is 25 less than the input squared.

Short answer

Question: Translate each verbal rule into a mathematical equation, expressing y in terms of x. a. The output is ten more than the input. b. The output is the sum of the input and 12 c. The output is five less than the input. d. The output is six times the input. e. The output is three less than six times the input. f. The output is six times the difference of the input and three. g. The output is 25 less than the input squared.

Step-by-step solution

a. The output is ten more than the input.

A direct translation of this rule into a mathematical equation would be: \(y = x + 10\)

b. The output is the sum of the input and 12

To translate this rule, we simply add x and 12: \(y = x + 12\)

c. The output is five less than the input.

This can be written as the difference between the input and 5: \(y = x - 5\)

d. The output is six times the input.

To represent this relationship, multiply the input by 6: \(y = 6x\)

e. The output is three less than six times the input.

First, we need to multiply the input by 6, and then subtract 3 from the result: \(y = 6x - 3\)

f. The output is six times the difference of the input and three.

First, we find the difference between the input and 3, and then multiply that difference by 6: \(y = 6(x - 3)\)

g. The output is 25 less than the input squared.

First, we square the input, and then subtract 25 from the result: \(y = x^2 - 25\)