Solving The Equation 7(x + 5) = 0 A Step-by-Step Guide
In the realm of mathematics, solving equations is a fundamental skill. This article will guide you through the process of solving a simple equation: 7(x + 5) = 0. We will break down each step, providing clear explanations and insights to help you grasp the underlying concepts. Whether you're a student learning algebra or simply brushing up on your math skills, this guide will equip you with the knowledge to tackle similar equations with confidence.
Understanding the Equation
At its core, an equation is a statement that asserts the equality of two expressions. In our case, the equation 7(x + 5) = 0 states that the expression on the left-hand side, 7(x + 5), is equal to the expression on the right-hand side, which is 0. The variable 'x' represents an unknown value that we aim to determine. To solve the equation means to find the value(s) of 'x' that make the equation true. In this case, we are dealing with a linear equation, meaning that the highest power of the variable 'x' is 1. Linear equations are generally straightforward to solve, and they have at most one solution.
Before diving into the solution process, it's important to understand the different parts of the equation. The number 7 is a coefficient, which is a numerical factor that multiplies a variable or an expression. The expression (x + 5) is a binomial, which is an algebraic expression consisting of two terms. The parentheses indicate that the entire binomial (x + 5) is multiplied by the coefficient 7. The number 0 on the right-hand side is a constant, which is a numerical value that does not change.
Understanding these components is crucial for applying the correct algebraic operations to isolate the variable 'x' and find its value. By carefully manipulating the equation, we can systematically eliminate terms and simplify the expression until we arrive at a solution for 'x'. The goal is to perform operations that maintain the equality of the equation while bringing us closer to isolating 'x' on one side.
Step-by-Step Solution
Now, let's embark on the journey of solving the equation 7(x + 5) = 0. We will proceed step-by-step, explaining the reasoning behind each action.
Step 1: Distribute the 7
The first step in solving this equation involves applying the distributive property. This property states that for any numbers a, b, and c, a(b + c) = ab + ac. In our case, we need to distribute the 7 across the binomial (x + 5).
Applying the distributive property, we multiply 7 by both terms inside the parentheses:
7 * x = 7x 7 * 5 = 35
Thus, the equation becomes:
7x + 35 = 0
This step is crucial because it removes the parentheses, making the equation easier to manipulate. By distributing the 7, we transform the equation into a simpler form that allows us to isolate the variable 'x'. This step is a fundamental technique in algebra and is often the first step in solving equations involving parentheses.
Step 2: Isolate the Term with x
Our next goal is to isolate the term containing 'x', which is 7x. To do this, we need to eliminate the constant term, which is 35, from the left-hand side of the equation. We can achieve this by subtracting 35 from both sides of the equation.
The principle behind this step is the addition property of equality, which states that if you add (or subtract) the same value from both sides of an equation, the equality remains true. By subtracting 35 from both sides, we maintain the balance of the equation while moving closer to isolating 'x'.
Subtracting 35 from both sides, we get:
7x + 35 - 35 = 0 - 35
Simplifying, we have:
7x = -35
Now, the term with 'x' is isolated on the left-hand side, and we have a simpler equation to solve.
Step 3: Solve for x
The final step is to solve for 'x' by isolating it completely. Currently, 'x' is being multiplied by 7. To undo this multiplication, we need to divide both sides of the equation by 7.
Again, we rely on the properties of equality. The division property of equality states that if you divide both sides of an equation by the same non-zero value, the equality remains true. Dividing both sides by 7 will isolate 'x' and give us the solution.
Dividing both sides by 7, we get:
(7x) / 7 = (-35) / 7
Simplifying, we find:
x = -5
Therefore, the solution to the equation 7(x + 5) = 0 is x = -5.
Verification of the Solution
To ensure that our solution is correct, we can substitute the value of x back into the original equation and check if it holds true. This process is called verification and is a crucial step in problem-solving.
Original equation: 7(x + 5) = 0
Substituting x = -5, we get:
7(-5 + 5) = 0
Simplifying the expression inside the parentheses:
7(0) = 0
Multiplying 7 by 0:
0 = 0
Since the left-hand side of the equation equals the right-hand side, the equation holds true. This confirms that our solution, x = -5, is indeed correct.
Verification is an essential step because it helps to catch any errors that may have occurred during the solution process. By substituting the solution back into the original equation, we can gain confidence in our answer and ensure that we have solved the problem correctly.
Conclusion
In this article, we have successfully solved the simple equation 7(x + 5) = 0. We started by understanding the equation and its components, then systematically applied algebraic operations to isolate the variable 'x'. We distributed the 7, subtracted 35 from both sides, and finally, divided both sides by 7 to arrive at the solution x = -5. We also verified our solution by substituting it back into the original equation.
Solving equations is a fundamental skill in mathematics, and mastering these techniques will enable you to tackle more complex problems. Remember to break down the problem into smaller steps, apply the properties of equality, and always verify your solution. With practice and patience, you can become proficient in solving equations and confidently navigate the world of algebra. This skill extends beyond academics, proving useful in various real-world scenarios where problem-solving and analytical thinking are essential.
Understanding the steps involved in solving the equation 7(x + 5) = 0 not only provides a solution but also reinforces foundational algebraic principles. The distributive property, addition property of equality, and division property of equality are cornerstone concepts that are applied across various mathematical disciplines. Grasping these concepts and their applications is crucial for building a strong foundation in mathematics. This solid grounding will pave the way for tackling more intricate mathematical problems, from complex equations to advanced calculus. By consistently practicing and applying these principles, you’ll develop a keen sense for mathematical problem-solving, enhancing your analytical skills and enabling you to approach challenges with greater confidence.
Moreover, the process of verifying the solution is just as important as the solution itself. Verification reinforces the understanding that a solution is not merely a numerical answer but a value that makes the equation true. This step cultivates a mindset of thoroughness and accuracy, which are invaluable in any analytical endeavor. By making verification a regular part of your problem-solving routine, you develop a critical eye for your own work, ensuring that your conclusions are well-supported and error-free. This methodical approach is a hallmark of mathematical proficiency and is highly transferable to other domains requiring precision and attention to detail. In essence, the ability to solve an equation like 7(x + 5) = 0 is more than just finding the value of 'x'; it is about mastering a process that embodies mathematical rigor and analytical thinking.
This methodical approach to problem-solving is not only beneficial in mathematics but also in various other fields. Whether it's analyzing data in a scientific experiment, making financial decisions, or even troubleshooting a technical issue, the ability to break down a complex problem into smaller, manageable steps is crucial for success. The skills honed in solving algebraic equations, such as attention to detail, logical reasoning, and the ability to identify and apply relevant principles, are highly transferable and can be applied in diverse contexts. Furthermore, the habit of verifying solutions fosters a critical mindset that is essential for accuracy and reliability in any field. In essence, mastering basic algebraic concepts like solving linear equations is an investment in developing a broader skillset that is valuable for academic pursuits, professional endeavors, and everyday problem-solving.
