Solving Equations A Step By Step Guide
In the realm of mathematics, solving equations is a fundamental skill that unlocks the doors to countless applications and deeper understanding. Equations, the mathematical sentences that assert the equality of two expressions, are the backbone of quantitative reasoning. Mastering the techniques to solve them is essential for anyone venturing into mathematics, science, engineering, or any field that relies on analytical thinking. This comprehensive guide delves into the intricacies of solving equations, providing step-by-step explanations, illustrative examples, and practical strategies to equip you with the proficiency to tackle a wide range of equation-solving challenges. From simple linear equations to more complex rational equations, this guide will empower you to approach equation-solving with confidence and clarity. Let's embark on this mathematical journey together, unraveling the mysteries of equations and discovering the satisfaction of finding their solutions.
1. Solving the Equation: $\frac{15}{x}=\frac{21}{3 x}+2$
Let's embark on a step-by-step journey to solve the equation $\frac{15}{x}=\frac{21}{3 x}+2$, unraveling its solution with clarity and precision. This equation, at first glance, may seem daunting, but with a systematic approach and a clear understanding of algebraic principles, we can conquer it. Our journey will involve a series of strategic steps, each designed to simplify the equation and bring us closer to the solution. We'll begin by identifying the common denominator, a crucial step in clearing the fractions and transforming the equation into a more manageable form. Then, we'll employ the distributive property, a powerful tool for expanding expressions and eliminating parentheses. Next, we'll skillfully combine like terms, streamlining the equation and paving the way for isolating the variable. Finally, we'll execute the crucial step of isolating the variable, the key to unlocking the solution and revealing the value of x that satisfies the equation. Throughout this process, we'll emphasize the importance of maintaining balance, ensuring that each operation performed on one side of the equation is mirrored on the other, preserving the fundamental equality that underpins the entire equation. With each step, we'll provide clear explanations and justifications, empowering you not just to find the solution, but to understand the underlying logic and principles that guide the process. So, let's dive in, and together, we'll unravel the solution to this equation, one step at a time.
Step 1: Identify the Common Denominator
The first step in tackling this equation is to identify the common denominator of the fractions involved. This crucial step will pave the way for clearing the fractions and transforming the equation into a more manageable form. By carefully examining the denominators, we can determine the least common multiple (LCM), which will serve as our common denominator. In this case, the denominators are x and 3x. The LCM of x and 3x is simply 3x. This is because 3x is a multiple of both x and itself. Identifying the common denominator is not just a procedural step; it's a strategic move that simplifies the equation and sets the stage for efficient solving. By eliminating the fractions, we transform the equation into a more familiar and easier-to-handle algebraic expression. So, with the common denominator in hand, we can proceed to the next step, confident that we're on the right track to finding the solution.
Step 2: Multiply Both Sides by the Common Denominator
Now that we've identified the common denominator as 3x, the next strategic move is to multiply both sides of the equation by this value. This crucial step is designed to eliminate the fractions, transforming the equation into a more manageable form that we can readily solve. When we multiply each term in the equation by 3x, we're essentially scaling up the entire equation, maintaining the balance and preserving the equality. This operation clears the fractions, allowing us to work with whole numbers and algebraic expressions, simplifying the subsequent steps in the solution process. By multiplying both sides by 3x, we're not just performing a mathematical operation; we're strategically maneuvering the equation towards a solution, paving the way for isolating the variable and unlocking its value. This step is a testament to the power of algebraic manipulation, demonstrating how we can transform complex equations into simpler, more solvable forms. So, with the fractions cleared, we're one step closer to the solution, and the path ahead is becoming clearer.
Step 3: Simplify the Equation
With the fractions now cleared, our equation has transformed into a more streamlined form: 45 = 21 + 6x. This simplified equation is a testament to the power of our previous steps, demonstrating how strategic algebraic manipulation can make complex problems more approachable. Our next task is to further simplify the equation by isolating the term containing the variable, 6x. To achieve this, we'll employ the principle of inverse operations, a fundamental concept in equation solving. We'll subtract 21 from both sides of the equation, effectively undoing the addition and moving us closer to isolating the variable. This step maintains the balance of the equation, ensuring that the equality remains intact. By subtracting 21 from both sides, we're not just performing a mathematical operation; we're strategically isolating the term containing the variable, paving the way for the final step of solving for x. This simplification is a crucial step in the process, bringing us closer to the solution with each operation. So, let's continue on this path, confident that we're making steady progress towards our goal.
Step 4: Isolate the Variable
Having simplified the equation to 24 = 6x, we've arrived at the final stage of our journey: isolating the variable x. This is the culmination of our efforts, the moment where we unlock the solution and reveal the value of x that satisfies the original equation. To isolate x, we'll once again employ the principle of inverse operations, this time focusing on undoing the multiplication. We'll divide both sides of the equation by 6, the coefficient of x, effectively separating x from its numerical multiplier. This division maintains the balance of the equation, ensuring that the equality remains intact. By dividing both sides by 6, we're not just performing a mathematical operation; we're strategically isolating x, bringing us to the final answer. This step is the key to solving the equation, the moment where the unknown becomes known. So, let's perform this final operation, and with a sense of accomplishment, unveil the solution.
Step 5: Verification
In the world of equation solving, verification is not just a formality; it's a crucial step that ensures the accuracy and validity of our solution. It's the mathematical equivalent of double-checking your work, a safeguard against errors and a confirmation that we've indeed found the correct value for the variable. To verify our solution, we'll substitute the value we obtained, x = 4, back into the original equation. This substitution transforms the equation into a numerical statement, which we can then evaluate to determine if it's true. If the left-hand side of the equation equals the right-hand side after the substitution, then our solution is verified, and we can confidently claim that x = 4 is indeed the correct answer. If, however, the two sides do not match, it signals that an error may have occurred during the solving process, prompting us to revisit our steps and identify the mistake. Verification is not just a procedural step; it's a cornerstone of mathematical rigor, a testament to the importance of precision and accuracy in our calculations. So, let's embrace the power of verification, and with a sense of thoroughness, confirm the validity of our solution.
Therefore, the solution to the equation is x = 4.
2. Solving the Equation: $\frac{11}{x^2+5 x+4}-\frac{3}{x+4}=\frac{1}{x+1}$
Let's embark on a step-by-step journey to solve the equation $\frac{11}{x^2+5 x+4}-\frac{3}{x+4}=\frac{1}{x+1}$, navigating its complexities with a clear and strategic approach. This equation, a blend of rational expressions, may appear intricate at first glance, but with a systematic methodology and a solid grasp of algebraic principles, we can unravel its solution. Our journey will involve a series of key steps, each designed to simplify the equation and bring us closer to the answer. We'll begin by factoring the quadratic expression in the denominator, a crucial step in identifying the common denominator and paving the way for clearing the fractions. Then, we'll skillfully combine the fractions, leveraging our understanding of fraction operations to simplify the equation. Next, we'll confront the equation head-on, solving for x using a variety of algebraic techniques. Finally, we'll engage in the critical step of checking for extraneous solutions, a safeguard against values that may satisfy the transformed equation but not the original. Throughout this process, we'll emphasize the importance of maintaining balance, ensuring that each operation performed on one side of the equation is mirrored on the other, preserving the fundamental equality that underpins the entire equation. With each step, we'll provide clear explanations and justifications, empowering you not just to find the solution, but to understand the underlying logic and principles that guide the process. So, let's dive in, and together, we'll unravel the solution to this equation, one step at a time.
Step 1: Factor the Quadratic Expression
The first crucial step in solving this equation is to factor the quadratic expression in the denominator, x² + 5x + 4. Factoring is a powerful technique that allows us to rewrite expressions in a more manageable form, often revealing hidden structures and simplifying subsequent operations. In this case, factoring the quadratic expression will help us identify the common denominator, a key element in clearing the fractions and transforming the equation into a more solvable form. To factor x² + 5x + 4, we seek two numbers that multiply to 4 and add up to 5. These numbers are 4 and 1. Therefore, we can rewrite the quadratic expression as (x + 4)(x + 1). Factoring is not just a mechanical process; it's a strategic move that simplifies the equation and sets the stage for efficient solving. By revealing the factors, we gain a deeper understanding of the expression's structure, which in turn facilitates the identification of the common denominator. So, with the quadratic expression factored, we're well-positioned to proceed to the next step, confident that we're on the right track to finding the solution.
Step 2: Identify the Common Denominator
With the quadratic expression factored as (x + 4)(x + 1), we've unlocked a crucial piece of the puzzle: the common denominator. Identifying the common denominator is a pivotal step in solving equations involving fractions, as it allows us to combine the fractions and eliminate them from the equation, transforming it into a more manageable form. By examining the denominators of the terms in the equation, we can determine the least common multiple (LCM), which will serve as our common denominator. In this case, the denominators are (x + 4)(x + 1), (x + 4), and (x + 1). The LCM of these expressions is simply (x + 4)(x + 1), as it encompasses all the factors present in the denominators. Identifying the common denominator is not just a procedural step; it's a strategic move that simplifies the equation and sets the stage for efficient solving. By recognizing the LCM, we can proceed to combine the fractions, paving the way for isolating the variable and unlocking its value. So, with the common denominator in hand, we can confidently move forward, knowing that we're well-equipped to tackle the next challenge.
Step 3: Multiply Both Sides by the Common Denominator
Now that we've identified the common denominator as (x + 4)(x + 1), the next strategic move is to multiply both sides of the equation by this expression. This crucial step is designed to eliminate the fractions, transforming the equation into a more manageable form that we can readily solve. When we multiply each term in the equation by (x + 4)(x + 1), we're essentially scaling up the entire equation, maintaining the balance and preserving the equality. This operation clears the fractions, allowing us to work with whole numbers and algebraic expressions, simplifying the subsequent steps in the solution process. By multiplying both sides by the common denominator, we're not just performing a mathematical operation; we're strategically maneuvering the equation towards a solution, paving the way for isolating the variable and unlocking its value. This step is a testament to the power of algebraic manipulation, demonstrating how we can transform complex equations into simpler, more solvable forms. So, with the fractions cleared, we're one step closer to the solution, and the path ahead is becoming clearer.
Step 4: Simplify and Solve for x
With the fractions now cleared, our equation has transformed into a more streamlined form: 11 - 3(x + 1) = 1(x + 4). This simplified equation is a testament to the power of our previous steps, demonstrating how strategic algebraic manipulation can make complex problems more approachable. Our next task is to further simplify the equation by expanding the expressions and combining like terms. To achieve this, we'll employ the distributive property, a fundamental concept in algebra that allows us to multiply a factor across a sum or difference. We'll distribute the -3 across (x + 1) and the 1 across (x + 4), effectively eliminating the parentheses and paving the way for combining like terms. Once we've expanded the expressions, we'll identify and combine like terms on each side of the equation, streamlining the equation and bringing us closer to isolating the variable x. This simplification is a crucial step in the process, making the equation more manageable and easier to solve. So, let's continue on this path, confident that we're making steady progress towards our goal.
Now, let's isolate x by performing the following operations:
Add 3x to both sides:
Subtract 4 from both sides:
Divide both sides by 4:
Step 5: Check for Extraneous Solutions
In the realm of solving rational equations, checking for extraneous solutions is not merely an optional step; it's an essential safeguard that ensures the validity of our solution. Extraneous solutions are values that emerge during the solving process but do not satisfy the original equation. These deceptive solutions often arise when we perform operations that can alter the domain of the equation, such as multiplying both sides by an expression containing the variable. To check for extraneous solutions, we must substitute the value we obtained, x = 1, back into the original equation. This substitution transforms the equation into a numerical statement, which we can then evaluate to determine if it's true. If the left-hand side of the equation equals the right-hand side after the substitution, then our solution is valid, and we can confidently claim that x = 1 is indeed a true solution. However, if the substitution leads to an undefined expression or a contradiction, it signals that the solution is extraneous and must be discarded. So, let's embrace the rigor of checking for extraneous solutions, and with a sense of thoroughness, ensure the validity of our answer.
Therefore, the solution to the equation is x = 1.
This comprehensive guide has walked you through the step-by-step process of solving two distinct equations, each with its own unique challenges and intricacies. By mastering the techniques and strategies outlined in this guide, you'll be well-equipped to tackle a wide range of equation-solving problems, unlocking the power of mathematics and applying it to various fields of study and real-world applications. So, embrace the challenge, and with confidence and clarity, continue your journey in the world of mathematics.
