Question

1. translate the following sentences into mathematical expressions or equations: a. one third of a number: ___________ b. the product of 6 and a number: _________ c. the sum of a number and 9 is equal to the product of that number and 3: ___________ 14. write each statement as an algebraic equation. use more than one variable and state what each variable represents. a. kim has 7 more than 5 times the amount sylvia has. b. the length of a rectangle is 3 feet less than 4 times its width.

Explanation:

a. Let the number be \( x \); one third of \( x \) is \( \frac{1}{3}x \).
b. Let the number be \( x \); product of 6 and \( x \) is \( 6x \).
c. Let the number be \( x \); sum of \( x \) and 9 equals product of \( x \) and 3: \( x + 9 = 3x \).
a. Let \( S \) = amount Sylvia has, \( K \) = amount Kim has; Kim has 7 more than 5 times Sylvia's amount: \( K = 5S + 7 \).
b. Let \( w \) = width (feet), \( l \) = length (feet); length is 3 less than 4 times width: \( l = 4w - 3 \).

Answer:

13a. \( \frac{1}{3}x \)
13b. \( 6x \)
13c. \( x + 9 = 3x \)
14a. \( K = 5S + 7 \), where \( S \) is the amount Sylvia has and \( K \) is the amount Kim has.
14b. \( l = 4w - 3 \), where \( w \) is the width (in feet) and \( l \) is the length (in feet).