Problem 18 - Introductory and Intermediate Algebra for College Students 4th

Question

Let \(x\) represent the number. Use the given conditions to write an equation. Solve the equation and find the number. Five more than four times a number is that number increased by \(35 .\) Find the number.

Step-by-Step Solution

Answer

The number is 10.

Step 1: Translate the Statement to Mathematical Equation

The problem can be translated into an equation as follows: 'five more than four times a number' refers to \(4x + 5\), and 'that number increased by 35' refers to \(x + 35\). Hence the equation to solve is: \(4x + 5 = x + 35\)

Step 2: Simplify the Equation

To make the variable \(x\) clear, isolate \(x\) on one side of the equation. Begin by subtracting \(x\) from both sides of the equation: \(4x - x = 35 - 5\). Simplify to get \(3x = 30\)

Step 3: Solve for x

Finally, to solve for \(x\), divide both sides of the equation by 3: \(x = 30/3\). Hence \(x = 10\).

Key Concepts

linear equations

Linear equations are a fundamental part of algebra. A linear equation is essentially an equation for a straight line. This is where the name 'linear' comes from. The equations have one or more variables with no exponents other than 1. In the equation we dealt with, both sides were linear expressions in terms of the variable.Understanding linear equations:

• They can be written in the form of \( ax + b = c \), where \( a \), \( b \), and \( c \) are constants.
• The highest degree of any variable in a linear equation is 1, which makes these equations straightforward to solve.

In our exercise, "four times a number plus five" and "the number plus 35" formed the two sides of a linear equation. This type of equation is straightforward to deal with once you recognize the structure.

problem solving

Problem solving with algebra requires translating verbal statements into mathematical expressions. This ability is key to tackling any real-world problem using mathematics. In our exercise, we had a problem statement describing a relationship between a number and some operations on it. Key steps to problem solving:

• Carefully read the problem and highlight key terms and numbers.
• Identify what you need to find, often marked as an unknown variable.
• Translate phrases into mathematical language (e.g., "five more than" means adding 5).
• Set up the equation and solve step-by-step.

Understanding these steps will not only help with current problems but equip you to handle varied math challenges in the future.

variable isolation

Isolating variables is a critical step in solving equations. When we talk about isolating a variable, we mean rearranging an equation so the variable stands alone on one side of the equation. This makes it easier to determine the exact value of the variable.How to isolate variables:

• Perform inverse operations, like subtraction or division, to move terms across the equal sign.
• Be consistent, do the same operation on both sides to maintain equality.
• Keep simplifying until the desired variable is isolated with a single coefficient (usually 1).

In our example, isolating the variable meant subtracting \( x \) from both sides and then dividing by the coefficient of \( x \), making it possible to find the value of \( x \) easily.