6(5n鈥?)鈥?0=10鈥?n+10

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30-03-2025

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This article will guide you step-by-step through solving the algebraic equation 6(5n – 7) – 30 = 10n + 10. We'll break down the process to make it easy to understand, even if you're new to algebra. Understanding how to solve this type of equation is crucial for many mathematical concepts.

Understanding the Equation

Before we begin, let's examine the equation: 6(5n – 7) – 30 = 10n + 10. This is a linear equation because the highest power of the variable 'n' is 1. Our goal is to isolate 'n' on one side of the equation to find its value.

Step-by-Step Solution

Here's how we'll solve the equation:

Step 1: Distribute the 6

The first step is to distribute the 6 to both terms inside the parentheses:

6 * 5n = 30n

6 * -7 = -42

Our equation now looks like this: 30n - 42 - 30 = 10n + 10

Step 2: Combine Like Terms

Next, combine the constant terms (-42 and -30) on the left side of the equation:

-42 - 30 = -72

The equation simplifies to: 30n - 72 = 10n + 10

Step 3: Isolate the Variable Term

Now, we need to get all the terms with 'n' on one side of the equation and the constant terms on the other. Let's subtract 10n from both sides:

30n - 10n - 72 = 10n - 10n + 10

This simplifies to: 20n - 72 = 10

Step 4: Isolate the Variable

Next, add 72 to both sides of the equation to isolate the term with 'n':

20n - 72 + 72 = 10 + 72

This gives us: 20n = 82

Step 5: Solve for n

Finally, divide both sides by 20 to solve for 'n':

20n / 20 = 82 / 20

This simplifies to: n = 4.1 or n = 82/20 = 41/10

Therefore, the solution to the equation 6(5n – 7) – 30 = 10n + 10 is n = 4.1

Verification

To verify our solution, substitute n = 4.1 back into the original equation:

6(5 * 4.1 – 7) – 30 = 10 * 4.1 + 10

6(20.5 – 7) – 30 = 41 + 10

6(13.5) – 30 = 51

81 – 30 = 51

51 = 51

The equation holds true, confirming our solution.

Conclusion

Solving algebraic equations like 6(5n – 7) – 30 = 10n + 10 involves a series of systematic steps. By following these steps carefully, you can successfully isolate the variable and find its value. Remember to always check your answer by substituting it back into the original equation. Practice is key to mastering this skill!