Problem 45 - Prealgebra

Question

Find each number. Three less than four times a number is \(-7 .\) Solve \(4 n-3=-7\)

Step-by-Step Solution

Answer

The number is -1.

Step 1: Understand the Equation

The problem statement gives the equation \(4n - 3 = -7\). Here, \(n\) is the variable representing the number we need to find.

Step 2: Isolate the Variable Term

Add 3 to both sides of the equation to eliminate the -3 on the left side:\[ 4n - 3 + 3 = -7 + 3 \]Simplifying both sides, we get:\[ 4n = -4 \]

Step 3: Solve for the Variable

Divide both sides of the equation by 4 to solve for \(n\):\[ \frac{4n}{4} = \frac{-4}{4} \]Which simplifies to:\[ n = -1 \]

Key Concepts

Solving Equations

When solving equations, you're finding the value of the variable that makes the equation true. Start by identifying your equation. In this case, it involves the number and operations that connect it to another value. Equations are like a seesaw: actions on one side must be mirrored by actions on the other to maintain balance. This ensures that whatever you do to one side, you do to the other.
Let's break it into steps:

• Manipulate the equation to isolate the variable. This typically involves reversing operations (e.g., addition cancels subtraction).
• For instance, if subtraction is present, add the same amount to both sides to cancel it out. Similarly, divide and conquer! If multiplication is used on the variable, divide both sides by the same number to isolate the variable.

After simplifying, you'll arrive at the solution.
Remember:

• Perform the same operation on both sides of the equation.
• Check your solution by substituting it back into the original equation to ensure it holds true.
• This approach is the essence of maintaining balance while solving equations.

Linear Equations

Linear equations in one variable are the simplest type of algebraic equations. They look like this: \[ ax + b = c \]where \(x\) is the variable, and \(a\), \(b\), and \(c\) are constants. The goal is to isolate the variable, \(x\), to find out exactly what it is.
The beauty of linear equations lies in their direct simplicity:

• They represent a straight line when graphed on a coordinate plane.
• They contain no exponents or powers - just plain addition, subtraction, multiplication, or division.

This straightforward nature means you can apply simple arithmetic operations to isolate \(x\). In our exercise, adding 3 and dividing by 4 were key steps in isolating \(n\).
Always remember:

• Operations performed across the equation must maintain the balance.
• Only basic arithmetic is necessary, highlighting the procedures you use daily.

Understanding how these equations work sets a strong foundation for tackling more complex algebraic problems.

Basic Algebra Concepts

Basic algebra is the foundation of mathematical problem solving. It introduces using symbols (like variables) to represent numbers. This allows us to express more general rules and relationships between numerical quantities.
In our example, the variable is \(n\), a placeholder for the number we want to find. The strength of algebra lies in its ability to:

• Abstractly represent and solve real-world problems.
• Simplify complex arithmetic into concise expressions with variables.
• Discover unknown quantities through manipulations.

Working with basic algebra requires using familiar operations in a slightly new context.
Key concepts include:

• Understanding variables: They are not fixed numbers but rather symbols standing in for unknown values.
• Performing operations: Similar to everyday arithmetic but with the added challenge of managing variables.
• Developing strategies: Techniques like balancing equations are critical for finding solutions.

Mastering these basic concepts prepares you for expanding knowledge and tackling the depth of algebraic challenges.