Problem 85 - Beginning and Intermediate Algebra

Question

Write each sentence as an equation or inequality. Use \(x\) to represent any unknown number. The sum of 5 and a number is 20.

Step-by-Step Solution

Answer

Equation: \(5 + x = 20\).

Step 1: Understanding the Problem

We need to write a sentence as an equation that involves a number and relate it to a given sum. The sentence given is 'The sum of 5 and a number is 20.'

Step 2: Identifying Keywords and Variables

The keyword 'sum' implies an addition operation. We can identify 'a number' as an unknown value, which we will represent with the variable \(x\).

Step 3: Setting Up the Equation

According to the sentence, when 5 is added to the unknown number \(x\), the result is 20. This relationship can be expressed as the equation: \(5 + x = 20\).

Step 4: Final Equation

The final equation representing the sentence 'The sum of 5 and a number is 20' is \(5 + x = 20\).

Key Concepts

Unknown Variables

In algebra, when we deal with unknown values, we often use variables to represent them. A variable is like a placeholder for a number that we do not yet know. In the context of the problem, the phrase "a number" refers to an unknown value.
To make solving equations easier, we assign this unknown a symbol, commonly, the letter "x." Using a variable allows us to set up an equation according to the information provided.

• It turns a problem that deals with an unknown into a mathematical form we can work with.
• Makes figuring out missing parts of a sum or product possible by providing a point of reference.

By establishing an unknown variable, we open the door to solving for its exact value through logical steps, such as in the provided exercise: 'The sum of 5 and a number is 20.' This lets us find out what that unknown number (x) is.

Addition Operation

Addition is one of the basic operations in mathematics, often signaling the combining of amounts. In equations, it is a straightforward tool used to relate components that add up to a total. The keyword 'sum' in our exercise signifies that addition is needed.
When we perform an addition operation, it involves combining two or more numbers. For instance, in the problem, to express 'The sum of 5 and a number is 20,'

• We first recognize the '+' symbol represents the addition operation.
• Each number involved in the addition is termed an 'addend,' such as the number 5 and the unknown variable x.

This operation allows us to see how two distinct parts (known and unknown) combine to form a total, in this case, reaching the sum of 20.

Algebraic Expressions

An algebraic expression is a combination of numbers, variables, and operators (such as +, -, ×, ÷). When we convert a sentence into an equation, we are creating an algebraic expression. This allows for the mathematical analysis needed to find unknown values.For example, in the sentence 'The sum of 5 and a number is 20,' we identify the components:

• The number 5, which is a known quantity.
• The unknown variable 'x,' representing the unknown number.
• The operation '+' indicating addition between these parts.

Thus, we form the algebraic expression: \(5 + x = 20\). By examining and solving this expression, we aim to determine the value of x that makes the equation true, effectively allowing us to solve the original problem.