Solving For X In The Equation 21/(x+3) - 1 = 2 A Step By Step Guide

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In this comprehensive guide, we will meticulously walk through the steps to determine the value of x in the given equation: 21x+3−1=2{ \frac{21}{x+3} - 1 = 2 }. This is a fundamental algebraic problem that involves fractions, variables, and equation-solving techniques. Understanding how to solve such equations is crucial for various mathematical applications and real-world problem-solving scenarios. Let's dive in and break down each step to ensure clarity and mastery of this concept.

Understanding the Equation

Before we jump into the solution, let's first understand the structure of the equation. We have a rational expression, 21x+3{ \frac{21}{x+3} }, which means 21 is divided by the expression x + 3. The equation also involves a constant term, -1, and another constant on the right side of the equation, 2. The goal is to isolate x and find its value that satisfies the equation. This involves performing algebraic manipulations to both sides of the equation while maintaining equality.

Key Algebraic Principles:

  1. Addition and Subtraction Property of Equality: You can add or subtract the same quantity from both sides of an equation without changing its solution.
  2. Multiplication and Division Property of Equality: You can multiply or divide both sides of an equation by the same non-zero quantity without changing its solution.
  3. Inverse Operations: Use inverse operations (addition/subtraction, multiplication/division) to isolate the variable.
  4. Combining Like Terms: Simplify each side of the equation by combining like terms.

By understanding these principles, we can systematically approach the problem and solve for x with confidence.

Step-by-Step Solution

Step 1: Isolate the Rational Expression

The first crucial step in solving the equation is to isolate the rational expression, 21x+3{ \frac{21}{x+3} }. To do this, we need to eliminate the constant term, -1, on the left side of the equation. We can achieve this by adding 1 to both sides of the equation. This maintains the equality and simplifies the equation.

Original equation:

21x+3−1=2{ \frac{21}{x+3} - 1 = 2 }

Add 1 to both sides:

21x+3−1+1=2+1{ \frac{21}{x+3} - 1 + 1 = 2 + 1 }

This simplifies to:

21x+3=3{ \frac{21}{x+3} = 3 }

Now we have successfully isolated the rational expression on one side of the equation. This sets us up for the next step, which involves dealing with the denominator.

Step 2: Eliminate the Denominator

To eliminate the denominator, which is x + 3, we need to multiply both sides of the equation by this expression. This will effectively cancel out the denominator on the left side and move us closer to isolating x. However, it's important to note that multiplying both sides by an expression containing x means we need to be mindful of potential extraneous solutions, which we will address later.

Multiply both sides by (x + 3):

21x+3×(x+3)=3×(x+3){ \frac{21}{x+3} \times (x+3) = 3 \times (x+3) }

On the left side, the (x + 3) terms cancel out, leaving us with:

21=3(x+3){ 21 = 3(x+3) }

This simplifies the equation and gets rid of the fraction, making it easier to solve for x. Now, we move on to distributing the 3 on the right side.

Step 3: Distribute and Simplify

Next, we need to distribute the 3 on the right side of the equation. This means multiplying 3 by both terms inside the parentheses, which are x and 3. This step is crucial for removing the parentheses and simplifying the equation further. Accurate distribution is essential to avoid errors in the solution.

Distribute 3 on the right side:

21=3×x+3×3{ 21 = 3 \times x + 3 \times 3 }

This simplifies to:

21=3x+9{ 21 = 3x + 9 }

Now, we have a linear equation that is much easier to solve. The next step involves isolating the term with x on one side of the equation.

Step 4: Isolate the Term with x

To isolate the term with x, which is 3x, we need to eliminate the constant term, 9, on the right side of the equation. We can do this by subtracting 9 from both sides. This maintains the balance of the equation and moves us closer to solving for x. Accurate subtraction is crucial to maintain the integrity of the equation.

Subtract 9 from both sides:

21−9=3x+9−9{ 21 - 9 = 3x + 9 - 9 }

This simplifies to:

12=3x{ 12 = 3x }

Now, we have a simplified equation with the term containing x almost isolated. The final step is to divide both sides by the coefficient of x.

Step 5: Solve for x

To finally solve for x, we need to divide both sides of the equation by the coefficient of x, which is 3. This will isolate x on one side of the equation and give us its value. Division is the final step in solving for x, and accurate division is essential to arrive at the correct answer.

Divide both sides by 3:

123=3x3{ \frac{12}{3} = \frac{3x}{3} }

This simplifies to:

4=x{ 4 = x }

Thus, the value of x that satisfies the equation is 4. However, it is essential to verify this solution to ensure it is not an extraneous solution.

Step 6: Verify the Solution

After finding a solution, it's crucial to verify it by substituting the value back into the original equation. This ensures that the solution satisfies the equation and is not extraneous. Extraneous solutions can arise when we multiply both sides of an equation by an expression containing x, as we did in Step 2. Verification is a critical step in problem-solving.

Original equation:

21x+3−1=2{ \frac{21}{x+3} - 1 = 2 }

Substitute x = 4:

214+3−1=2{ \frac{21}{4+3} - 1 = 2 }

Simplify the denominator:

217−1=2{ \frac{21}{7} - 1 = 2 }

Divide 21 by 7:

3−1=2{ 3 - 1 = 2 }

Simplify:

2=2{ 2 = 2 }

The left side of the equation equals the right side, which confirms that x = 4 is a valid solution.

Final Answer

Therefore, the value of x in the equation 21x+3−1=2{ \frac{21}{x+3} - 1 = 2 } is 4. We have successfully solved the equation by isolating the rational expression, eliminating the denominator, distributing and simplifying, isolating the term with x, solving for x, and verifying the solution. This step-by-step process ensures a clear and accurate understanding of the solution.

In summary, the solution to the equation 21x+3−1=2{ \frac{21}{x+3} - 1 = 2 } is x = 4. This was achieved through a series of algebraic manipulations, including isolating the rational expression, eliminating the denominator, distributing terms, and verifying the solution to ensure its validity.

Common Mistakes to Avoid

When solving equations like this, there are several common mistakes students often make. Being aware of these mistakes can help you avoid them and improve your problem-solving accuracy.

  1. Incorrect Distribution: A common mistake is to not distribute properly when multiplying a number by an expression in parentheses. For example, in Step 3, failing to multiply both x and 3 by 3 would lead to an incorrect equation.
  2. Arithmetic Errors: Simple arithmetic errors, such as adding or subtracting incorrectly, can lead to a wrong answer. It's crucial to double-check each step to avoid such mistakes.
  3. Forgetting to Verify: As mentioned earlier, forgetting to verify the solution can lead to accepting extraneous solutions. Always substitute your solution back into the original equation to ensure its validity.
  4. Incorrectly Eliminating the Denominator: When multiplying both sides by the denominator, ensure you multiply every term on both sides, not just the terms involving the fraction.
  5. Sign Errors: Sign errors, particularly when dealing with negative numbers, are common. Pay close attention to the signs of the terms and perform operations carefully.

By being mindful of these common mistakes and practicing regularly, you can improve your equation-solving skills and achieve greater accuracy.

Practice Problems

To further enhance your understanding and skills in solving equations with rational expressions, here are a few practice problems:

  1. Solve for x: 15x−2+3=5{ \frac{15}{x-2} + 3 = 5 }
  2. Solve for y: 10y+1−2=1{ \frac{10}{y+1} - 2 = 1 }
  3. Solve for z: 8z−3+4=6{ \frac{8}{z-3} + 4 = 6 }

Working through these practice problems will solidify your understanding of the steps involved and help you become more confident in solving similar equations. Remember to follow the same steps we outlined in this guide: isolate the rational expression, eliminate the denominator, distribute and simplify, isolate the variable term, solve for the variable, and verify your solution.

By mastering these techniques, you'll be well-equipped to tackle a wide range of algebraic problems involving rational expressions and equations. Keep practicing, and you'll see significant improvement in your problem-solving abilities!