identify the terms, like terms, coefficients, and constants in each expression. (example 1)
1. 4e + 7e + 5 + 2e
2. 5a + 2 + 7 + 6a
3. 4 + 4y + y + 3

for each verbal phrase, define a variable to represent the unknown quantity. then write the phrase as an algebraic expression. (examples 2–4)
4. three more pancakes than hector ate
5. twelve fewer questions than were on the first test
6. two and one - half times the number of minutes spent exercising
7. one - third the number of yards
8. four less than seven times lynns age
9. $2.50 more than one - fourth the cost of a pizza
10. a plumber charges $50 to visit a house plus $40 for every hour of work. define a variable to represent the unknown quantity. then write an expression to represent the total cost of hiring a plumber. (example 4)
11. a gymnastics studio charges an annual fee of $35 plus $20 per class. define a variable to represent the unknown quantity. then write an expression to represent the total cost of taking classes. (example 4)
12. a rectangle has a length that is half its width. define a variable to represent the unknown quantity. then write an expression to represent the perimeter of the rectangle. (example 5)
13. in a triangle there are two sides that have the same length and the third side is 1.5 times longer than the length of the other two. define a variable to represent the unknown quantity. then write an algebraic expression to represent the perimeter of the triangle. (example 5)

Question

identify the terms, like terms, coefficients, and constants in each expression. (example 1)
1. 4e + 7e + 5 + 2e
2. 5a + 2 + 7 + 6a
3. 4 + 4y + y + 3

for each verbal phrase, define a variable to represent the unknown quantity. then write the phrase as an algebraic expression. (examples 2–4)
4. three more pancakes than hector ate
5. twelve fewer questions than were on the first test
6. two and one - half times the number of minutes spent exercising
7. one - third the number of yards
8. four less than seven times lynns age
9. $2.50 more than one - fourth the cost of a pizza
10. a plumber charges $50 to visit a house plus $40 for every hour of work. define a variable to represent the unknown quantity. then write an expression to represent the total cost of hiring a plumber. (example 4)
11. a gymnastics studio charges an annual fee of $35 plus $20 per class. define a variable to represent the unknown quantity. then write an expression to represent the total cost of taking classes. (example 4)
12. a rectangle has a length that is half its width. define a variable to represent the unknown quantity. then write an expression to represent the perimeter of the rectangle. (example 5)
13. in a triangle there are two sides that have the same length and the third side is 1.5 times longer than the length of the other two. define a variable to represent the unknown quantity. then write an algebraic expression to represent the perimeter of the triangle. (example 5)

Accepted Answer

Let's solve question 4 as an example:

# Explanation:
## Step1: Define the variable
Let $ h $ represent the number of pancakes Hector ate.
## Step2: Write the algebraic expression
"Three more pancakes than Hector ate" means we add 3 to the number of pancakes Hector ate. So the algebraic expression is $ h + 3 $.

# Answer:
Variable: Let $ h $ = number of pancakes Hector ate.
Algebraic expression: $ h + 3 $

For question 5:

# Explanation:
## Step1: Define the variable
Let $ q $ represent the number of questions on the first test.
## Step2: Write the algebraic expression
"Twelve fewer questions than were on the first test" means we subtract 12 from the number of questions on the first test. So the algebraic expression is $ q - 12 $.

# Answer:
Variable: Let $ q $ = number of questions on the first test.
Algebraic expression: $ q - 12 $

For question 6:

# Explanation:
## Step1: Define the variable
Let $ m $ represent the number of minutes spent exercising.
## Step2: Write the algebraic expression
"Two and one - half times the number of minutes spent exercising" means we multiply the number of minutes spent exercising by $ 2\frac{1}{2} $ (or $ \frac{5}{2} $). So the algebraic expression is $ \frac{5}{2}m $ (or $ 2.5m $).

# Answer:
Variable: Let $ m $ = number of minutes spent exercising.
Algebraic expression: $ \frac{5}{2}m $ (or $ 2.5m $)

For question 7:

# Explanation:
## Step1: Define the variable
Let $ y $ represent the number of yards.
## Step2: Write the algebraic expression
"One - third the number of yards" means we multiply the number of yards by $ \frac{1}{3} $. So the algebraic expression is $ \frac{1}{3}y $.

# Answer:
Variable: Let $ y $ = number of yards.
Algebraic expression: $ \frac{1}{3}y $

For question 8:

# Explanation:
## Step1: Define the variable
Let $ a $ represent Lynn's age.
## Step2: Write the algebraic expression
"Seven times Lynn's age" is $ 7a $, and "four less than seven times Lynn's age" means we subtract 4 from $ 7a $. So the algebraic expression is $ 7a - 4 $.

# Answer:
Variable: Let $ a $ = Lynn's age.
Algebraic expression: $ 7a - 4 $

For question 9:

# Explanation:
## Step1: Define the variable
Let $ c $ represent the cost of a pizza.
## Step2: Write the algebraic expression
"One - fourth the cost of a pizza" is $ \frac{1}{4}c $, and "$ \$2.50 $ more than one - fourth the cost of a pizza" means we add 2.50 to $ \frac{1}{4}c $. So the algebraic expression is $ \frac{1}{4}c+ 2.50 $.

# Answer:
Variable: Let $ c $ = cost of a pizza.
Algebraic expression: $ \frac{1}{4}c + 2.50 $

For question 10:

# Explanation:
## Step1: Define the variable
Let $ h $ represent the number of hours the plumber works.
## Step2: Write the algebraic expression
The plumber charges $ \$50 $ to visit (a constant) plus $ \$40 $ for every hour of work. So the total cost $ C $ is the sum of the fixed cost ($ 50 $) and the variable cost ($ 40h $). The algebraic expression is $ C=50 + 40h $.

# Answer:
Variable: Let $ h $ = number of hours the plumber works.
Algebraic expression: $ 50 + 40h $

For question 11:

# Explanation:
## Step1: Define the variable
Let $ n $ represent the number of classes.
## Step2: Write the algebraic expression
The gymnastics studio charges an annual fee of $ \$35 $ (a constant) plus $ \$20 $ per class. So the total cost $ C $ is the sum of the annual fee ($ 35 $) and the cost per class times the number of classes ($ 20n $). The algebraic expression is $ C = 35+20n $.

# Answer:
Variable: Let $ n $ = number of classes.
Algebraic expression: $ 35 + 20n $

For question 12:

# Explanation:
## Step1: Define the variable
Let $ w $ represent the width of the rectangle. Then the length $ l=\frac{1}{2}w $ (since the length is half the width).
## Step2: Recall the perimeter formula for a rectangle
The perimeter $ P $ of a rectangle is given by $ P = 2(l + w) $. Substitute $ l=\frac{1}{2}w $ into the formula: $ P=2(\frac{1}{2}w+w) $. Simplify the expression inside the parentheses: $ \frac{1}{2}w + w=\frac{1 + 2}{2}w=\frac{3}{2}w $. Then $ P = 2\times\frac{3}{2}w=3w $.

# Answer:
Variable: Let $ w $ = width of the rectangle.
Algebraic expression for perimeter: $ 3w $ (or by expanding $ 2(\frac{1}{2}w + w)=2\times\frac{1}{2}w+2w=w + 2w = 3w $)

For question 13:

# Explanation:
## Step1: Define the variable
Let $ s $ represent the length of the two equal - length sides of the triangle. Then the length of the third side is $ 1.5s $.
## Step2: Recall the perimeter formula for a triangle
The perimeter $ P $ of a triangle is the sum of the lengths of its three sides. So $ P=s + s+1.5s $. Combine like terms: $ s + s+1.5s=(1 + 1+1.5)s = 3.5s $ (or $ \frac{7}{2}s $).

# Answer:
Variable: Let $ s $ = length of the two equal - length sides.
Algebraic expression for perimeter: $ 3.5s $ (or $ \frac{7}{2}s $ or $ s + s+1.5s $)

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Question

Explanation:

Step1: Define the variable

Step2: Write the algebraic expression

Step1: Define the variable

Step2: Write the algebraic expression

Step1: Define the variable

Step2: Write the algebraic expression

Answer: