Solving For X In The Equation 9(-9x + 5) + 10 = 2(-10x + 2)

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Introduction

In this comprehensive guide, we will delve into the process of solving for x in the given equation: 9(-9x + 5) + 10 = 2(-10x + 2). This is a fundamental algebraic problem that requires a step-by-step approach to arrive at the simplest form of the solution. Understanding how to solve such equations is crucial for various mathematical and real-world applications. This article will provide a detailed explanation of each step, ensuring clarity and ease of understanding. We will cover the initial distribution, simplification, combining like terms, isolating the variable x, and finally, arriving at the solution in its simplest form.

Step-by-Step Solution

1. Distribute the Constants

The first step in solving the equation is to distribute the constants outside the parentheses to the terms inside. This involves multiplying the numbers preceding the parentheses with each term within the parentheses. This process helps in eliminating the parentheses and simplifies the equation.

  • For the left side of the equation, we distribute 9 to both -9x and 5:
    • 9 * (-9x) = -81x
    • 9 * 5 = 45 So, 9(-9x + 5) becomes -81x + 45.
  • For the right side of the equation, we distribute 2 to both -10x and 2:
    • 2 * (-10x) = -20x
    • 2 * 2 = 4 So, 2(-10x + 2) becomes -20x + 4.

After distributing the constants, the equation transforms to:

-81x + 45 + 10 = -20x + 4

2. Simplify Both Sides

After distributing, the next step is to simplify both sides of the equation by combining like terms. This involves adding or subtracting constant terms on each side to condense the equation further. Simplifying makes the equation easier to work with and helps in isolating the variable.

  • On the left side, we combine the constants 45 and 10:
    • 45 + 10 = 55 So, -81x + 45 + 10 simplifies to -81x + 55.
  • The right side of the equation, -20x + 4, is already in its simplest form as there are no like terms to combine.

Thus, the simplified equation is:

-81x + 55 = -20x + 4

3. Move Variables to One Side

The next crucial step is to move all terms containing the variable x to one side of the equation and the constants to the other side. This is achieved by adding or subtracting terms from both sides of the equation while maintaining the equality. The goal is to isolate x on one side, which will eventually lead to the solution.

  • To move the variable terms, we can add 81x to both sides of the equation. This will eliminate the -81x term on the left side:
    • -81x + 55 + 81x = -20x + 4 + 81x
    • This simplifies to 55 = 61x + 4.
  • Now, we have the variable term on the right side, and the constant term on the left.

4. Isolate the Variable Term

Now that we have all the x terms on one side, the next step is to isolate the variable term. This means getting the term with x by itself on one side of the equation. To do this, we need to eliminate any constants that are added to or subtracted from the variable term. We accomplish this by performing the opposite operation on both sides of the equation.

  • In our equation, 55 = 61x + 4, we have a constant 4 added to the term 61x. To isolate 61x, we subtract 4 from both sides:
    • 55 - 4 = 61x + 4 - 4
    • This simplifies to 51 = 61x.

5. Solve for x

The final step in solving for x is to divide both sides of the equation by the coefficient of x. The coefficient is the number multiplied by x. Dividing by the coefficient will isolate x and give us the solution.

  • In our equation, 51 = 61x, the coefficient of x is 61. To solve for x, we divide both sides by 61:
    • 51 / 61 = 61x / 61
    • This gives us x = 51/61.

6. Simplify the Fraction (If Possible)

The final step is to check if the solution can be simplified. A fraction is in its simplest form when the numerator and the denominator have no common factors other than 1. To simplify a fraction, we look for the greatest common divisor (GCD) of the numerator and the denominator and divide both by it.

  • In our case, the fraction is 51/61. The number 51 can be factored into 3 * 17, and 61 is a prime number, meaning it is only divisible by 1 and itself. Since 51 and 61 have no common factors other than 1, the fraction is already in its simplest form.

Therefore, the simplest form of the solution for x is:

x = 51/61

Verification

To ensure that our solution is correct, we can substitute the value of x back into the original equation and verify that both sides of the equation are equal. This process helps in identifying any errors made during the solving process.

Original equation:

9(-9x + 5) + 10 = 2(-10x + 2)

Substitute x = 51/61:

9(-9(51/61) + 5) + 10 = 2(-10(51/61) + 2)

First, we calculate the expressions inside the parentheses:

  • -9(51/61) = -459/61
  • -459/61 + 5 = -459/61 + (5 * 61)/61 = -459/61 + 305/61 = -154/61
  • -10(51/61) = -510/61
  • -510/61 + 2 = -510/61 + (2 * 61)/61 = -510/61 + 122/61 = -388/61

Now, substitute these values back into the equation:

9(-154/61) + 10 = 2(-388/61)

Perform the multiplication:

  • 9 * (-154/61) = -1386/61
  • 2 * (-388/61) = -776/61

Substitute these results back into the equation:

-1386/61 + 10 = -776/61

Convert 10 to a fraction with a denominator of 61:

  • 10 = (10 * 61)/61 = 610/61

Substitute this value back into the equation:

-1386/61 + 610/61 = -776/61

Combine the fractions on the left side:

(-1386 + 610)/61 = -776/61

-776/61 = -776/61

Since both sides of the equation are equal, our solution x = 51/61 is correct.

Common Mistakes to Avoid

When solving linear equations, several common mistakes can lead to incorrect solutions. Being aware of these pitfalls can help in avoiding them and ensuring accuracy.

  1. Incorrect Distribution: One of the most common errors is not distributing constants correctly across terms within parentheses. Remember to multiply the constant by each term inside the parentheses.
  2. Combining Unlike Terms: Only like terms (terms with the same variable and exponent, or constants) can be combined. Mixing variable terms with constants is a frequent mistake.
  3. Sign Errors: Pay close attention to signs (positive and negative) when adding, subtracting, multiplying, and dividing. A small sign error can change the entire solution.
  4. Not Performing Operations on Both Sides: To maintain equality, any operation performed on one side of the equation must also be performed on the other side. Failing to do so will lead to an unbalanced equation and an incorrect solution.
  5. Incorrect Order of Operations: Follow the correct order of operations (PEMDAS/BODMAS) to ensure accurate calculations. Parentheses, Exponents, Multiplication and Division, Addition and Subtraction.
  6. Forgetting to Simplify: Always simplify the final answer as much as possible. This includes reducing fractions to their simplest form and combining like terms.

Conclusion

Solving linear equations for x is a fundamental skill in algebra. By following a systematic approach, we can accurately find the value of x. In this article, we have demonstrated a step-by-step method to solve the equation 9(-9x + 5) + 10 = 2(-10x + 2). We started by distributing constants, simplified both sides of the equation, moved variables to one side, isolated the variable term, and finally solved for x. The solution we found, x = 51/61, was verified by substituting it back into the original equation.

Understanding and avoiding common mistakes is also crucial for solving equations accurately. Paying attention to distribution, combining like terms, sign errors, and maintaining balance in the equation can significantly improve your problem-solving skills.

By mastering these techniques, you will be well-equipped to tackle more complex algebraic problems and real-world applications. Keep practicing and reinforcing these skills to build a strong foundation in mathematics.