Problem 5 - Elementary Algebra

Question

Solve. $$12 x+1=5$$

Step-by-Step Solution

Answer

\(x = \frac{1}{3}\)

Step 1 - Isolate the Term with the Variable

To solve the equation \(12x + 1 = 5\), we first need to isolate the term containing the variable \(x\). To do this, subtract 1 from both sides of the equation to keep it balanced. This will give us:\[12x + 1 - 1 = 5 - 1\]Simplify the equation:\[12x = 4\]

Step 2 - Solve for x

Now that we have isolated the term with \(x\), we need to get \(x\) by itself. To do this, divide both sides of the equation by 12. So we have:\[\frac{12x}{12} = \frac{4}{12}\]Simplify the division:\[x = \frac{1}{3}\]

Key Concepts

Solving Equations

Solving equations is a fundamental skill in algebra. When we solve an equation, we are finding the value of the variable that makes the equation true. In our given problem, the equation is \(12x + 1 = 5\). This means that when we substitute the correct number for \(x\), both sides of the equation will be equal. To solve an equation, there are important steps to follow:

• Ensure the equation is correctly set with all terms in place.
• Watch out for any constants and terms involving the variable.
• Proceed step by step to simplify or rearrange accordingly.

By following these steps, we make sure to handle each component of the equation and find the correct solution.

Isolation of Variables

Isolation of variables is a key concept when solving linear equations. This means that our goal is to get the variable (in this case, \(x\)) by itself on one side of the equation. For \(12x + 1 = 5\), we start by isolating the variable term \(12x\).To achieve this, we perform operations on both sides of the equation:

• Subtract the constant term from both sides to remove it from the equation. For example, subtracting 1 gives us \(12x = 4\).
• This subtraction is crucial because it maintains the balance of the equation, a fundamental rule in algebra.

Once the variable is isolated, it sets the stage for the final step: solving for the actual value.

Simplification

Simplification is all about making an equation easier to understand and solve. This involves reducing the equation to its simplest form. After isolating the variable term, \(12x = 4\), we need to simplify further to find \(x\).Here's how simplification works in this context:

• Division is used to simplify the expression. Divide both sides by 12 to maintain equality. This step directly simplifies \(12x = 4\) to \(x = \frac{4}{12}\).
• Further reduce \(\frac{4}{12}\) to its simplest fractional form: \(x = \frac{1}{3}\).

Simplification ensures that we present the solution clearly, making it easy to understand and verify.