Problem 60 - Elementary Algebra

Question

For the following problems, translate the following phrases or sentences into mathematical expressions or equations. Seven is added to ten less than some number. The result is one

Step-by-Step Solution

Answer

Answer: x - 3 = 1

Step 1: Identify the unknown number

Let's use a variable, x, to represent the unknown number.

Step 2: Express "ten less than some number" in terms of x

Ten less than x can be represented as x - 10.

Step 3: Express "seven is added to ten less than some number" in terms of x

Seven is added to x - 10, so the expression is (x - 10) + 7.

Step 4: Write an equation representing "The result is one" using the expression

The result of the expression in Step 3 is equal to one. This can be represented as an equation: (x - 10) + 7 = 1.

Step 5: Simplify the equation as needed

Combine like terms on the left-hand side of the equation to simplify: x - 3 = 1.

Key Concepts

Translating Phrases into Equations

Translating phrases into equations is a crucial skill in algebra that allows us to convert word problems into mathematical expressions. By doing so, we can solve problems more easily. In our example, the phrase "Seven is added to ten less than some number" needs to be translated into an equation.
First, identify the key components:

• "some number" refers to the unknown variable, which we'll call \( x \).
• "ten less than some number" translates to \( x - 10 \).
• "Seven is added to" indicates addition to this expression, resulting in \( (x - 10) + 7 \).
• "The result is one" tells us that this expression equals 1.

By carefully translating each part of the phrase, we can write the equation: \( (x - 10) + 7 = 1 \). This equation now represents the problem in mathematical form.

Algebraic Expressions

An algebraic expression is a combination of numbers, variables, and operations like addition, subtraction, multiplication, and division. These expressions represent quantities and can be used to model real-world situations.
In our particular example, we first encounter the algebraic expression \( x - 10 \). This expression denotes 'ten less than some number.' By itself, \( x \) is a simple variable, and altering it by subtracting 10 makes it an expression.
When we further add 7 to this expression, it becomes \( (x - 10) + 7 \). Here, parentheses are used to clarify the order of operations, ensuring the subtraction occurs before the addition. This algebraic expression represents a part of the real-world situation described in the problem statement.

Unknown Variables

An unknown variable is a symbol, often \( x \), used to represent a number that we don't know yet. It acts as a placeholder, allowing us to perform operations and solve equations.
In the exercise given, the unknown variable is represented by \( x \). We know there is a number that, when reduced by 10 and then increased by 7, results in a total of 1. The purpose of using \( x \) is to help us find this elusive number through algebraic manipulation.
By incorporating this unknown variable into our equation, we form \( (x - 10) + 7 = 1 \), which we can then solve to determine the value of \( x \), achieving a deeper understanding of the problem context.

Simplifying Equations

Simplifying equations involves combining like terms and reducing an equation to its most basic form, making it easier to analyze or solve. This is a critical step in algebra, especially when dealing with more complex expressions.
In the provided solution, the expression \( (x - 10) + 7 \) needs simplifying. To do this, combine like terms, specifically the constant terms \(-10\) and \(7\). By performing the arithmetic, we get \( x - 3 \), a simpler expression.
The complete simplified equation is \( x - 3 = 1 \). By having a simplified expression, solving for \( x \) becomes straightforward, allowing us to efficiently find the unknown number that satisfies the original problem statement.