Problem 11 - College Algebra and Calculus: An Applied Approach

Question

In Exercises 11-16, write an equation that represents the statement. The sum of 5 and \(x\) equals 8 .

Step-by-Step Solution

Answer

The equation that represents the statement 'The sum of 5 and x equals 8' is \(5 + x = 8\).

Step 1: Identify Variables

From the statement, it's clear that 'x' is the variable that we're trying to find.

Step 2: Construct Equation

Given the 'sum of 5 and x equals 8', it translates to '5 plus x equals 8', which can be written in mathematical form as \(5 + x = 8\).

Step 3: Simplification

Simplify the equation if needed. However, in this case, our equation \(5 + x = 8\) is in it's simplest form, so no further simplification is required.

Key Concepts

Variables

In algebraic equations, variables are essential components. They are symbols that represent unknown values. In our exercise, the variable is represented by the letter 'x'.
Variables can be letters or symbols used frequently in equations to show that the value is unknown or can change. They allow flexibility because variables can take on different values depending on the equation.

• 'x' might be a common choice, but variables can be any letter or symbol.
• Variables can represent numbers, but sometimes they stand for more complex things, like sets or vectors, in advanced math.
• The role of the variable is to balance the equation, meaning it transforms as necessary to maintain equality.

Understanding variables is crucial as they are fundamental to creating mathematical expressions and solving equations.

Equation Construction

Constructing an equation involves translating a given statement into a mathematical sentence. This is a critical step when solving problems in math. In the example we have, the statement says, "The sum of 5 and _x_ equals 8." Breaking it down involves taking the verbal expression and turning it into symbols and numbers.

• 'Sum' indicates addition.
• The number 5 is one part of our sum.
• 'X' is the second part, signifying the unknown value we are trying to find.
• Finally, 'equals 8' sets up the equation balance, indicating the result of the addition.

This results in the equation: \(5 + x = 8\). This process of constructing equations from statements is foundational, as it applies to many areas of mathematics and real-life situations.

Mathematical Form

The mathematical form of an equation is how it is neatly written using mathematical symbols and numbers. This form is important for clarity and precision. In our case, the equation \(5 + x = 8\) clearly represents the problem statement in symbolic language. Writing in mathematical form involves following specific rules:

• Start with the known values and operations: here, '5 +' signifies addition with the variable 'x'.
• Use the equal sign '=' to show that both sides of the equation are balanced.
• Ensure the equation is as simple as possible, which means there are no unnecessary symbols or steps.

The beauty of the mathematical form is its ability to convey complex ideas succinctly. It allows anyone familiar with the basic rules of algebra to understand and interpret the statement accurately.