Representing the First Number

Oct 13, 2025 by miodeya.com 98 views

In this article, we will guide Nancy through the process of solving a math problem that involves finding the product of two numbers. The problem requires her to use algebraic expressions and apply mathematical operations to arrive at the correct solution.

Breaking Down the Problem

The problem states that Nancy needs to find the product of two numbers. The first number is represented as 14 minus $x$ , where $x$ is a variable. The second number is 19 more than 7 times the first number. To solve this problem, Nancy must first understand the relationship between the two numbers and then apply mathematical operations to find their product.

Representing the First Number

The first number is represented as 14 minus $x$ . This can be written as an algebraic expression: $14 - x$ . The variable $x$ represents an unknown value that needs to be determined in order to find the value of the first number.

Representing the Second Number

The second number is 19 more than 7 times the first number. This can be written as an algebraic expression: $7(14 - x) + 19$ . The expression $7(14 - x)$ represents 7 times the first number, and the expression $+ 19$ represents 19 more than the result.

Finding the Product

To find the product of the two numbers, Nancy must multiply the two expressions together. This can be written as:

$(14 - x) \times (7(14 - x) + 19)$

To simplify this expression, Nancy can use the distributive property of multiplication over addition. This allows her to multiply each term in the first expression by each term in the second expression.

Simplifying the Expression

Using the distributive property, Nancy can simplify the expression as follows:

$(14 - x) \times (7(14 - x) + 19) = (14 - x) \times 98 - (14 - x) \times x + 19(14 - x)$

This expression can be further simplified by combining like terms.

Combining Like Terms

The expression can be simplified by combining like terms as follows:

$(14 - x) \times 98 - (14 - x) \times x + 19(14 - x) = 98(14 - x) - x(14 - x) + 19(14 - x)$

This expression can be further simplified by combining like terms.

Final Simplification

The expression can be simplified by combining like terms as follows:

$98(14 - x) - x(14 - x) + 19(14 - x) = 1372 - 98x - 14x + x^2 + 266 - 19x$

This expression can be further simplified by combining like terms.

Final Answer

The final simplified expression is:

$1638 - 130x + x^2$

This is the product of the two numbers, and it is expressed in terms of the variable $x$ .

Conclusion

In the previous article, we guided Nancy through the process of solving a math problem that involved finding the product of two numbers. In this article, we will answer some frequently asked questions related to this topic.

Q: What is the product of two numbers?

A: The product of two numbers is the result of multiplying the two numbers together. In the case of the problem we solved earlier, the product of the two numbers is represented by the expression $1638 - 130x + x^2$ .

Q: How do I represent the product of two numbers algebraically?

A: To represent the product of two numbers algebraically, you can use the distributive property of multiplication over addition. This allows you to multiply each term in the first expression by each term in the second expression.

Q: What is the distributive property of multiplication over addition?

A: The distributive property of multiplication over addition is a mathematical property that allows you to multiply each term in an expression by each term in another expression. This property is represented by the following equation:

$a(b + c) = ab + ac$

Q: How do I simplify an algebraic expression?

A: To simplify an algebraic expression, you can use various mathematical operations such as combining like terms, factoring, and canceling out common factors.

Q: What are like terms?

A: Like terms are terms that have the same variable or variables raised to the same power. For example, the terms $2x$ and $5x$ are like terms because they both have the variable $x$ raised to the power of 1.

Q: How do I combine like terms?

A: To combine like terms, you can add or subtract the coefficients of the like terms. For example, if you have the expression $2x + 5x$ , you can combine the like terms by adding the coefficients: $2x + 5x = 7x$ .

Q: What is factoring?

A: Factoring is the process of expressing an algebraic expression as a product of simpler expressions. For example, the expression $6x$ can be factored as $2 \times 3x$ .

Q: How do I factor an algebraic expression?

A: To factor an algebraic expression, you can look for common factors among the terms. For example, if you have the expression $6x + 12$ , you can factor out the common factor $6$ : $6x + 12 = 6(x + 2)$ .

Q: What is canceling out common factors?

A: Canceling out common factors is the process of eliminating common factors from an algebraic expression. For example, if you have the expression $6x + 12$ , you can cancel out the common factor $6$ : $6x + 12 = 6(x + 2)$ .

Conclusion

In this article, we answered some frequently asked questions related to the product of two numbers. We discussed the distributive property of multiplication over addition, simplifying algebraic expressions, combining like terms, factoring, and canceling out common factors. We hope that this article has provided you with a better understanding of these mathematical concepts.