Problem 52 - Prealgebra

Question

Solve each equation. Check your solution. $$n+4.7=13.6$$

Step-by-Step Solution

Answer

The solution is \(n = 8.9\).

Step 1: Isolate the Variable

To solve the equation \(n + 4.7 = 13.6\), we need to isolate \(n\) on one side. We can do this by subtracting 4.7 from both sides of the equation. This gives us:\[ n + 4.7 - 4.7 = 13.6 - 4.7 \]Simplifying both sides we get:\[ n = 8.9 \]

Step 2: Verify the Solution

To ensure that our solution is correct, we substitute \(n = 8.9\) back into the original equation.Start with the original equation:\[ n + 4.7 = 13.6 \]Substitute \(8.9\) for \(n\):\[ 8.9 + 4.7 \]Calculate:\[ 8.9 + 4.7 = 13.6 \]Since both sides of the original equation are equal, our solution for \(n\) is verified as correct.

Key Concepts

Isolate the Variable

One of the most important initial steps when solving an equation is isolating the variable. This essentially means getting the variable on one side of the equation all by itself. Let's take a closer look, using the example equation: \(n + 4.7 = 13.6\). The variable here is \(n\).
To isolate \(n\), you need to get rid of the number added to it, which in this case is 4.7. This is done by performing the opposite operation. Since 4.7 is added to \(n\), we subtract 4.7 from both sides of the equation. Here's why:

• Subtract 4.7 from both sides: \(n + 4.7 - 4.7 = 13.6 - 4.7\).
• Simplify both sides to maintain the balance of the equation: \(n = 8.9\).

By carrying out these steps, \(n\) becomes isolated, resulting in \(n = 8.9\). This transformation allows us to understand precisely the value \(n\) must take to satisfy the original equation. Being consistent with this approach is a key piece of solving equations.

Verify Solution

Once you think you've found the solution to an equation, verifying the solution is a crucial step to ensure accuracy. This involves substituting your solution back into the original equation to check if it holds true. For example, with our solved equation \(n = 8.9\), we need to substitute \(n\) back into the original equation \(n + 4.7 = 13.6\).
Here’s how verification works:

• Replace \(n\) with 8.9 and calculate: \(8.9 + 4.7\).
• By doing the math, you get \(13.6 = 13.6\).

Since both sides of the equation are equal, we can confidently say our solution is correct. Verification helps prevent common mistakes and reinforces the correctness of the solution. Always include this step when solving equations to strengthen your math skills.

Substitution Method

The substitution method is a practical technique used not only to verify solutions but also to solve systems of equations. While our example primarily uses it for verification, let's explore its broader purpose.
Substitution involves replacing a variable with a given value or expression. This aids in simplification and helps solve equations more easily. In our context, after isolating \(n = 8.9\), substituting back into the. initial equation \(n + 4.7 = 13.6\) confirms the equation holds true:

• Substitute 8.9 into \(n + 4.7\), yielding \(13.6\).
• The outcome \(13.6 = 13.6\) validates our solution.

More broadly, the substitution method allows you to break down complex equations into simpler ones by expressing one variable in terms of others. It's a versatile strategy often applied in algebra and real-world problem-solving scenarios, highlighting its importance in mathematical toolkits.