Problem 78 - Algebra 1: Concepts and Skills

Question

Solve the equation. $$ x+1=6 $$

Step-by-Step Solution

Answer

The solution to the equation is x = 5.

Step 1: Rewrite the Equation

The equation is given as \(x + 1 = 6\). The goal is to solve for x, which means getting x on its own on one side of the equation.

Step 2: Isolate the Variable

To remove the 1 from the left side of the equation and isolate x, subtract 1 from both sides of the equation. This is demonstrated by the equation \(x + 1 -1 = 6 - 1\), which simplifies to \(x = 5\).

Step 3: Present the Solution

Given that \(x = 5\), the solution to the equation is x = 5.

Key Concepts

Solving Equations

Solving equations forms the foundation of algebra and involves finding the value of a variable that makes an equation true. In our exercise, we have the equation \( x + 1 = 6 \). Solving equations often requires performing operations that keep the equation balanced. This means whatever operation you do to one side, you must do to the other as well. This ensures that the equality remains true throughout the process.

Here’s how we solve our given equation:

• Identify the variable you need to solve for, which is \( x \) in this case.
• Our goal is to have \( x \) alone on one side of the equation so we can determine its value.
• We need to eliminate any numbers or operations that are beside the variable, often by performing inverse operations.

Once you isolate the variable, you substitute back into the original equation to verify your solution. For the equation \( x + 1 = 6 \), subtracting 1 from both sides gives \( x = 5 \). You can check this by substituting 5 back into the original equation: \( 5 + 1 = 6 \), which is correct.

Algebraic Manipulation

Algebraic manipulation is the process of rearranging and simplifying equations, expressions, or formulas through a variety of techniques. This includes adding, subtracting, multiplying, or dividing terms, as well as applying properties of equality. These skills are crucial for solving equations effectively.

In solving \( x + 1 = 6 \), algebraic manipulation involves simplifying the equation by using inverse operations. Inverse operations are operations that undo each other, like addition and subtraction or multiplication and division.

• In our equation, we have "+1" with our variable \( x \). To isolate \( x \), we perform the inverse operation of addition, which is subtraction.
• We subtract 1 from both sides, simplifying the equation to find \( x = 5 \).
• This process is fundamental: always perform operations equally on both sides to maintain balance.

Mastering basic techniques of algebraic manipulation will help in solving various types of equations, preparing you for more complex problems.

Basic Algebra Concepts

Understanding basic algebra concepts is key when you start solving equations. Algebra helps us express relationships and changes using symbols and letters, enabling us to solve equations and find unknown values.

In the equation \( x + 1 = 6 \), some fundamental algebra concepts are at play:

• Variables: Letters like \( x \) represent unknown quantities. These are what you solve for in an equation.
• Constants: The numbers that stand alone, such as 1 and 6 in our equation, which do not change.
• Operations: Addition, subtraction, multiplication, and division show relationships between different terms. Here, addition is the operation being used.

A good understanding of these basic concepts helps build confidence and skills for dealing with algebraic equations. The relationship between variables and constants, along with proper manipulation skills, allows you to find solutions efficiently. Remember, algebra is like solving a puzzle; once you understand how the pieces fit together, finding the solution becomes much easier.