Problem 43 - Prealgebra

Question

Find each number. Five more than twice a number is 27 . Solve \(2 n+5=27\).

Step-by-Step Solution

Answer

The number is 11.

Step 1: Understand the Equation

The equation given is \(2n + 5 = 27\). The problem states that five more than twice a number equals 27. Our goal is to find the value of \(n\).

Step 2: Isolate the Variable

To solve for \(n\), we first need to get rid of the constant 5 by subtracting 5 from both sides of the equation: \(2n + 5 - 5 = 27 - 5\), simplifying to \(2n = 22\).

Step 3: Solve for the Variable

Now, divide both sides of the equation by 2 to solve for \(n\): \(\dfrac{2n}{2} = \dfrac{22}{2}\), which simplifies to \(n = 11\).

Step 4: Verify the Solution

To ensure the solution is correct, substitute \(n = 11\) back into the original equation: \(2(11) + 5 = 22 + 5 = 27\). This confirms that our solution is correct.

Key Concepts

Equation Simplification

When dealing with linear equations like \(2n + 5 = 27\), the first step is often equation simplification. This involves transforming the equation into a more manageable form. The aim is to gradually remove any additional numbers or terms that are clinging to the variable "\(n\)."
One way to start is by removing constants from one side of the equation by doing the same mathematical operation to both sides.
This keeps the equation balanced.Take a look at our example. We have "5 more than twice a number" – so, \(2n + 5\). To simplify, we subtract 5 from both sides:

• This action is often referred to as "undoing" – we undo the addition of 5 with a subtraction.
• It leaves us with \(2n = 22\) on one side of the equation.

Now, the equation is simpler, with the constant (5) eliminated, making it easier to solve in subsequent steps.

Variable Isolation

After you've simplified the equation, it's time to focus on variable isolation. The goal here is to get "\(n\)" by itself on one side of the equation. This often requires performing an operation opposite to the one currently binding the variable.
In our problem, \(2n = 22\) is the simplified equation. To isolate \(n\):

• Focus on the "2" that is attached to \(n\). It's currently multiplying \(n\).
• Divide both sides by 2 to "undo" the multiplication, thus isolating \(n\).

When you divide, the equation becomes \(n = 11\). Now we have successfully isolated the variable. This process is crucial because it reveals the value of the unknown in the equation.

Verification of Solutions

Once you've found a solution, like \(n = 11\) in our exercise, it's vital to verify it. Verification ensures that the value you obtained truly satisfies the original equation.
Testing the solution means substituting the value back in the original equation to check if both sides remain equal. For this example:

• Substitute \(n = 11\) into the initial equation \(2n + 5 = 27\).
• The left side becomes \(2 \times 11 + 5\) which simplifies to \(22 + 5\).
• This equals \(27\), perfectly matching the right side.

Because the equation remains balanced, the solution is confirmed correct. Verification is an important step to solidify your understanding and ensure no errors were made in the solving process.