Problem 40 - Intermediate Algebra

Question

Solve each equation. Check each result. See Example 4. $$ 9 n+36=6 n $$

Step-by-Step Solution

Answer

The solution to the equation is \(n = -12\).

Step 1: Understand the Equation

We start with the equation \(9n + 36 = 6n\). Our goal is to solve for \(n\), which means finding the value of \(n\) that makes this equation true.

Step 2: Move Variables to One Side

Subtract \(6n\) from both sides of the equation to get all terms involving \(n\) on one side: \[ 9n + 36 - 6n = 6n - 6n \] Simplifying gives: \[ 3n + 36 = 0 \]

Step 3: Isolate the Variable

Subtract 36 from both sides to isolate the term containing \(n\): \[ 3n + 36 - 36 = 0 - 36 \] So we have: \[ 3n = -36 \]

Step 4: Solve for n

Divide both sides by 3 to solve for \(n\): \[ \frac{3n}{3} = \frac{-36}{3} \] Simplifying gives: \[ n = -12 \]

Step 5: Verify the Solution

Substitute \(n = -12\) back into the original equation to check if it is correct: \[ 9(-12) + 36 = 6(-12) \] Calculate each side: \[ -108 + 36 = -72 \] \[ -72 = -72 \] Since both sides are equal, \(n = -12\) is the correct solution.

Key Concepts

Solving Equations

Solving equations is a fundamental skill in algebra that involves finding the value of a variable that makes the equation true. The equation we are focusing on is \[9n + 36 = 6n\] This simply means that the expression on the left side is equal to the expression on the right side. To solve such equations, you typically aim to "undo" the operations around the variable until you can clearly see what the variable equals. Think of it as peeling away the layers around the variable to reveal its value. The process often involves removing constants and coefficients through simple arithmetic operations like addition, subtraction, multiplication, or division.

For our example:

• Subtract 6n from both sides to bring all the terms with \(n\) on one side.
• Subtract constants to further simplify the equation.
• Divide to isolate the variable.

By following these steps, you will identify the value of \(n\) that satisfies the equation.

Variable Isolation

Variable isolation is the heart of solving equations. It involves getting the variable by itself on one side of the equation so that you can easily determine its value. In our equation, we started with \[3n + 36 = 0\] Our goal was to isolate \(n\). Here's how we achieve it:

• Subtract 36 from both sides to remove it and maintain balance. Doing so gives us \(3n = -36\).
• Divide both sides by the coefficient in front of \(n\), which in this case is 3. This helps us finally isolate \(n\).

This results in \(n = -12\). The key here is performing equal operations on both sides of the equation to keep it balanced. By gradually peeling away the numbers and coefficients that enclose \(n\), you finally have well-isolated variables.

Checking Solutions

Checking solutions is a crucial final step in solving linear equations. It's a way to ensure that the value you found indeed satisfies the original equation. In our example: \[n = -12\] Once we have this result, we substitute \(n = -12\) back into the original equation \(9n + 36 = 6n\) to verify:- Substitute \(n = -12\): \(9(-12) + 36\) and \(6(-12)\)- Simplify both sides: - The left side becomes \(-108 + 36 = -72\) - The right side becomes \(-72\)

Both sides of the equation match, thus confirming that \(n = -12\) is indeed correct. This step is especially important for avoiding errors, ensuring that the solution is robust and reliable. Always remember: verifying your solution builds confidence and ensures accuracy.