Problem 15 - Elementary and Intermediate Algebra

Question

Solve each equation. Check the result. $$ 4 x+1=13 $$

Step-by-Step Solution

Answer

The solution is \(x = 3\).

Step 1: Isolate the variable term

Subtract 1 from both sides of the equation to get the variable term by itself. This will start to isolate the variable term on one side.\[4x + 1 - 1 = 13 - 1\]This simplifies to:\[4x = 12\]

Step 2: Solve for the variable

Divide both sides of the equation by 4 to solve for the variable \(x\).\[\frac{4x}{4} = \frac{12}{4}\]This simplifies to:\[x = 3\]

Step 3: Check the solution

Substitute \(x = 3\) back into the original equation to ensure it satisfies the equation:\[4(3) + 1 = 13\]Calculate the left side:\[12 + 1 = 13\]This confirms that the solution \(x = 3\) is correct.

Key Concepts

Solving Linear Equations

When solving linear equations, the overall goal is to find the value of the variable that makes the equation true. A linear equation is typically in the form ax + b = c. In our exercise, we are working with the equation \(4x + 1 = 13\).
To solve this equation, the first step is to isolate the variable term. This means we need to get \(x\) by itself on one side of the equation. As highlighted in the solution:

• Subtract 1 from both sides to remove the constant on the side with \(x\). This gives us \(4x = 12\).

Once the variable term is isolated, the next focus is finding the exact value of \(x\). This involves unraveling the coefficient of \(x\) by dividing both sides of the equation by 4.

• This results in \(x = 3\), which is the solution.

The core idea here is to perform operations that simplify the equation and bring the variable into focus, making it straightforward to identify its value.

Algebraic Manipulation

Algebraic manipulation is a powerful mathematical tool used to rearrange equations and expressions to highlight the variable of interest. It involves performing operations that keep the equation balanced but also gradually simplify it. When we manipulate equations, we are using basic operations like:

• Adding or subtracting numbers from both sides
• Multiplying or dividing each side by a non-zero number

In our example, we begin by subtracting 1 from both sides. This reduces the constant terms, which is an algebraic manipulation practice aimed at simplifying the equation. Next, division is applied to both sides to solve for \(x\) efficiently.
Through algebraic manipulation, we ensure that we are not changing the equality of the equation but instead making it easier to read and analyze. This technique helps keep equations logical and balanced at each step, paving the way for finding solutions.

Checking Solutions

After solving an equation, it is crucial to verify that the solution is correct. Checking solutions is about ensuring that what you found truly satisfies the original equation. In our situation, after determining \(x = 3\), we substitute it back into the original equation \(4x + 1 = 13\). This step is called the substitution method.
By replacing \(x\) with 3, the equation becomes \(4(3) + 1 = 13\). We then multiply and add to check if the calculation aligns with 13, the original right side of the equation. By confirming that both sides of the equation equal after substitution, we are assured that our solution is valid.
This verification process not only boosts confidence in the solved answer but also helps identify errors early on if the results do not match. Always remember, no equation is fully solved until the solution has been checked for accuracy.