Problem 69 - Beginning and Intermediate Algebra

Question

One more than twice a number is 5 .

Step-by-Step Solution

Answer

The number is 2.

Step 1: Define the variable

Let the unknown number be represented by the variable \( x \).

Step 2: Set up the equation

According to the problem, one more than twice a number is 5. This can be written as the equation: \( 2x + 1 = 5 \).

Step 3: Solve for the variable

Subtract 1 from both sides of the equation to isolate the term with \( x \): \( 2x + 1 - 1 = 5 - 1 \), which simplifies to \( 2x = 4 \).

Step 4: Isolate the variable

Divide both sides of the equation by 2 to solve for \( x \): \( \frac{2x}{2} = \frac{4}{2} \), yielding \( x = 2 \).

Step 5: Verify the solution

Substitute \( x = 2 \) back into the original equation to ensure it satisfies the condition: \( 2(2) + 1 = 4 + 1 = 5 \). So, the solution is correct.

Key Concepts

Defining Variables

When we solve linear equations, the first step is to define the variable. A variable represents an unknown number. It is often denoted by letters like x, y, or z. In this exercise, we use x to represent the unknown number. By defining the unknown as x, we can translate verbal statements into mathematical equations. This process is essential for breaking down and solving problems systematically.

Setting Up Equations

Once we have defined our variable, the next step is to set up the equation. This involves converting the words of the problem into a mathematical statement. The original problem 'One more than twice a number is 5' can be written as:
• 'Twice a number' translates to \(2x\).
• 'One more than twice a number' means we add 1 to \(2x\), which gives us \(2x + 1\).
• Finally, 'is 5' translates to an equation: \(2x + 1 = 5\).
Setting up the equation correctly is crucial because it forms the basis for solving the problem.

Isolating Variables

To solve the equation, we need to isolate the variable. This means we need to get x by itself on one side of the equation. Here’s how we do it step-by-step:
• Start with the equation: \(2x + 1 = 5\).
• Subtract 1 from both sides to get rid of the 1 added to 2x: \(2x + 1 - 1 = 5 - 1\), which simplifies to \(2x = 4\).
• Next, divide both sides by 2 to isolate x: \(\frac{2x}{2} = \frac{4}{2}\), which simplifies to \(x = 2\).
By isolating the variable, we find that \(x = 2\). This tells us that the unknown number in the problem is 2.

Verifying Solutions

The final step in solving linear equations is to verify the solution. This ensures our solution is correct. We substitute \(x = 2\) back into the original equation:
• Original equation: \(2x + 1 = 5\).
• Substitute \(x = 2\): \(2(2) + 1 = 5\).
• Simplify: \(4 + 1 = 5\), which is true.
Since both sides of the equation are equal, we have verified our solution is correct. Verifying solutions is a critical step to make sure that we didn’t make any mistakes in our calculations.