Answer :
Absolutely! Let's explore how we can translate verbal expressions into algebraic expressions by following the table provided.
### Step-by-Step Solution
1. Understanding Variables:
- When we see phrases like "a number," "some number," or "the number," we use variables such as [tex]\( x, y, z, a, b, \)[/tex] or [tex]\( c \)[/tex]. These variables represent unknown quantities.
2. Expressions Involving Addition (More Than, Greater Than):
- Verbal Expression: "the sum of 3 and a number"
- Algebraic Expression: [tex]\( 3 + a \)[/tex]
- Explanation: Here, 3 is a constant and [tex]\(a\)[/tex] is the variable. The operation indicated is addition.
3. Expressions Involving Subtraction (Less Than):
- Verbal Expression: "a number less than 5"
- Algebraic Expression: [tex]\( 5 - y \)[/tex]
- Explanation: In this context, 5 is a constant, and [tex]\(y\)[/tex] is the variable. The operation indicated is subtraction, meaning [tex]\(y\)[/tex] is being subtracted from 5.
4. Expressions Involving Multiplication (Product Of):
- Verbal Expression: "the product of 12 and a number"
- Algebraic Expression: [tex]\( 12r \)[/tex]
- Explanation: Here, 12 is a coefficient, and [tex]\(r\)[/tex] is the variable. The operation indicated is multiplication.
5. Expressions Involving Division (Quotient Of, By):
- Verbal Expression: "the quotient of a number and 7"
- Algebraic Expression: [tex]\( \frac{q}{7} \)[/tex]
- Explanation: The variable [tex]\(q\)[/tex] is divided by the constant 7.
6. Expressions Involving Parentheses (The Quantity Of):
- Verbal Expression: "4 times the quantity of [tex]\(a\)[/tex] plus 2"
- Algebraic Expression: [tex]\( 4(a + 2) \)[/tex]
- Explanation: Parentheses are used to indicate that we first add [tex]\(a\)[/tex] and 2, and then multiply the result by 4.
7. Expressions Involving Exponents (Power Of, Raised To):
- Verbal Expression: "7 raised to a number"
- Algebraic Expression: [tex]\( 7^x \)[/tex]
- Explanation: This indicates an exponential operation where 7 is the base and [tex]\(x\)[/tex] is the exponent.
### Practice Problems for Translating Verbal to Algebraic Expressions:
1. The sum of 9 and a number:
- Verbal: "the sum of 9 and a number"
- Algebraic: [tex]\( 9 + b \)[/tex]
- Explanation: 9 is the constant and [tex]\( b \)[/tex] is the variable.
2. A number decreased by 3:
- Verbal: "a number decreased by 3"
- Algebraic: [tex]\( x - 3 \)[/tex]
- Explanation: [tex]\( x \)[/tex] is the variable and 3 is the constant.
3. 4 times the sum of a number and 5:
- Verbal: "4 times the sum of a number and 5"
- Algebraic: [tex]\( 4(n + 5) \)[/tex]
- Explanation: Parentheses group [tex]\( n + 5 \)[/tex] together and it is multiplied by 4.
4. The quotient of a number and 10:
- Verbal: "the quotient of a number and 10"
- Algebraic: [tex]\( \frac{k}{10} \)[/tex]
- Explanation: [tex]\( k \)[/tex] is the variable and it is divided by the constant 10.
By following and understanding these basic principles and examples, you should be able to translate verbal mathematical phrases into their corresponding algebraic expressions accurately. Keep practicing with different phrases to become more proficient in this skill!
### Step-by-Step Solution
1. Understanding Variables:
- When we see phrases like "a number," "some number," or "the number," we use variables such as [tex]\( x, y, z, a, b, \)[/tex] or [tex]\( c \)[/tex]. These variables represent unknown quantities.
2. Expressions Involving Addition (More Than, Greater Than):
- Verbal Expression: "the sum of 3 and a number"
- Algebraic Expression: [tex]\( 3 + a \)[/tex]
- Explanation: Here, 3 is a constant and [tex]\(a\)[/tex] is the variable. The operation indicated is addition.
3. Expressions Involving Subtraction (Less Than):
- Verbal Expression: "a number less than 5"
- Algebraic Expression: [tex]\( 5 - y \)[/tex]
- Explanation: In this context, 5 is a constant, and [tex]\(y\)[/tex] is the variable. The operation indicated is subtraction, meaning [tex]\(y\)[/tex] is being subtracted from 5.
4. Expressions Involving Multiplication (Product Of):
- Verbal Expression: "the product of 12 and a number"
- Algebraic Expression: [tex]\( 12r \)[/tex]
- Explanation: Here, 12 is a coefficient, and [tex]\(r\)[/tex] is the variable. The operation indicated is multiplication.
5. Expressions Involving Division (Quotient Of, By):
- Verbal Expression: "the quotient of a number and 7"
- Algebraic Expression: [tex]\( \frac{q}{7} \)[/tex]
- Explanation: The variable [tex]\(q\)[/tex] is divided by the constant 7.
6. Expressions Involving Parentheses (The Quantity Of):
- Verbal Expression: "4 times the quantity of [tex]\(a\)[/tex] plus 2"
- Algebraic Expression: [tex]\( 4(a + 2) \)[/tex]
- Explanation: Parentheses are used to indicate that we first add [tex]\(a\)[/tex] and 2, and then multiply the result by 4.
7. Expressions Involving Exponents (Power Of, Raised To):
- Verbal Expression: "7 raised to a number"
- Algebraic Expression: [tex]\( 7^x \)[/tex]
- Explanation: This indicates an exponential operation where 7 is the base and [tex]\(x\)[/tex] is the exponent.
### Practice Problems for Translating Verbal to Algebraic Expressions:
1. The sum of 9 and a number:
- Verbal: "the sum of 9 and a number"
- Algebraic: [tex]\( 9 + b \)[/tex]
- Explanation: 9 is the constant and [tex]\( b \)[/tex] is the variable.
2. A number decreased by 3:
- Verbal: "a number decreased by 3"
- Algebraic: [tex]\( x - 3 \)[/tex]
- Explanation: [tex]\( x \)[/tex] is the variable and 3 is the constant.
3. 4 times the sum of a number and 5:
- Verbal: "4 times the sum of a number and 5"
- Algebraic: [tex]\( 4(n + 5) \)[/tex]
- Explanation: Parentheses group [tex]\( n + 5 \)[/tex] together and it is multiplied by 4.
4. The quotient of a number and 10:
- Verbal: "the quotient of a number and 10"
- Algebraic: [tex]\( \frac{k}{10} \)[/tex]
- Explanation: [tex]\( k \)[/tex] is the variable and it is divided by the constant 10.
By following and understanding these basic principles and examples, you should be able to translate verbal mathematical phrases into their corresponding algebraic expressions accurately. Keep practicing with different phrases to become more proficient in this skill!
