Solving -5(3-7x) = 9x + 11 A Step-by-Step Guide

In the realm of mathematics, linear equations are fundamental building blocks. These equations, characterized by variables raised to the first power, represent straight lines when graphed. Mastering the art of solving linear equations is crucial for success in algebra and beyond. In this comprehensive guide, we will dissect the linear equation -5(3-7x) = 9x + 11, step-by-step, providing a clear pathway to the solution. This equation may seem daunting at first glance, but with a systematic approach, it becomes manageable and even enjoyable to solve. Understanding the properties of equality and the order of operations are key concepts that we will utilize throughout this process. Before diving into the specifics of this equation, it's essential to grasp the broader concept of what a linear equation represents and why solving it is significant.

Linear equations are not just abstract mathematical constructs; they are powerful tools for modeling real-world phenomena. From calculating distances and speeds to predicting financial trends, linear equations provide a framework for understanding and solving problems across various disciplines. The ability to manipulate and solve these equations is a skill that translates directly into practical applications. Moreover, the logical and analytical thinking developed through solving linear equations forms a strong foundation for tackling more complex mathematical problems in the future. Whether you're a student just beginning your algebraic journey or someone looking to refresh your skills, this guide aims to provide a thorough and accessible explanation of the process involved in solving linear equations, using the example equation as a central case study. By the end of this guide, you will not only understand how to solve this particular equation but also gain a deeper appreciation for the elegance and utility of linear equations in mathematics and beyond.

The first critical step in solving the linear equation -5(3-7x) = 9x + 11 involves applying the distributive property. This property is a cornerstone of algebraic manipulation, allowing us to simplify expressions by multiplying a term outside parentheses with each term inside. In our equation, the term -5 is multiplied with both 3 and -7x. Let's break down this process:

• Multiplying -5 by 3: -5 multiplied by 3 equals -15. This operation sets the stage for simplifying the left side of the equation.
• Multiplying -5 by -7x: A negative number multiplied by another negative number results in a positive number. Therefore, -5 multiplied by -7x equals +35x. This step is crucial as it introduces the variable term on the left side of the equation, which we will later isolate.

By applying the distributive property, we transform the original equation into a more manageable form: -15 + 35x = 9x + 11. This simplified equation now presents us with a clearer path towards isolating the variable 'x'. Distributing correctly ensures that each term within the parentheses is properly accounted for, laying a solid foundation for the subsequent steps in solving the equation. The distributive property is not just a mechanical rule; it's a fundamental concept that reflects the structure of mathematical operations. Understanding and applying it correctly is essential for solving not only linear equations but also a wide range of algebraic problems. By carefully executing this initial step, we pave the way for a smooth and accurate solution.

After applying the distributive property, our equation stands as -15 + 35x = 9x + 11. The next strategic move is to consolidate the terms involving 'x' on one side of the equation. This process, often referred to as combining like terms, simplifies the equation and brings us closer to isolating the variable 'x'.

To achieve this, we need to eliminate the 'x' term from one side of the equation. A common approach is to subtract the smaller 'x' term from both sides. In our case, 9x is smaller than 35x. So, we subtract 9x from both the left and right sides of the equation:

-15 + 35x - 9x = 9x + 11 - 9x

This operation maintains the balance of the equation, a critical principle in algebra. By subtracting the same term from both sides, we ensure that the equality remains valid. On the left side, 35x - 9x simplifies to 26x. On the right side, 9x - 9x cancels out, leaving us with just 11. Our equation now looks like this:

-15 + 26x = 11

This step has successfully grouped the 'x' terms on one side, making the equation significantly simpler. Combining like terms is a fundamental technique in algebra, and mastering it is crucial for solving various types of equations. By strategically moving terms, we streamline the equation and make it easier to isolate the variable. This step not only simplifies the equation but also demonstrates the power of algebraic manipulation in solving complex problems.

Having consolidated the 'x' terms, our equation now reads -15 + 26x = 11. The subsequent goal is to isolate the variable 'x' further. This involves moving the constant terms (the numbers without 'x') to the opposite side of the equation. In our case, we need to eliminate the -15 from the left side.

To do this, we perform the inverse operation. Since -15 is being subtracted, we add 15 to both sides of the equation. This maintains the balance of the equation, a principle we've emphasized throughout our solution process:

-15 + 26x + 15 = 11 + 15

Adding 15 to both sides effectively cancels out the -15 on the left side. On the left, -15 + 15 becomes 0, leaving us with just 26x. On the right side, 11 + 15 equals 26. Our equation now simplifies to:

26x = 26

This step has successfully isolated the term containing 'x' on one side of the equation. By strategically adding the constant term, we have moved closer to our ultimate goal of finding the value of 'x'. Isolating the variable is a core technique in solving equations, and it relies on the principle of performing inverse operations to maintain equality. This step showcases the power of algebraic manipulation in simplifying equations and bringing them closer to a solution.

At this stage, our equation is 26x = 26. We are now in the final stretch of solving for 'x'. The variable 'x' is currently being multiplied by 26. To isolate 'x' completely, we need to perform the inverse operation: division. We divide both sides of the equation by 26:

26x / 26 = 26 / 26

This step is crucial as it directly leads to the value of 'x'. On the left side, 26x divided by 26 simplifies to x, as the 26s cancel each other out. On the right side, 26 divided by 26 equals 1. Therefore, our equation now reads:

x = 1

This is the solution to the linear equation. We have successfully isolated 'x' and determined its value. Dividing both sides by the coefficient of 'x' is the final step in solving many linear equations. It's a direct and effective way to unveil the value of the variable. This step underscores the importance of inverse operations in algebra and demonstrates how we can systematically manipulate equations to arrive at a solution. By performing this final division, we have completed the process and found the value of 'x' that satisfies the original equation.

After meticulously applying the principles of algebraic manipulation, we have arrived at the solution to the linear equation -5(3-7x) = 9x + 11. Our step-by-step journey, involving distribution, combining like terms, and isolating the variable, has led us to the answer:

x = 1

This result signifies the value of 'x' that, when substituted back into the original equation, will make the equation true. It is the unique value that satisfies the equality. This final answer is not just a number; it represents the culmination of a logical and systematic process. It underscores the power of algebra in solving problems and revealing hidden values.

To ensure the accuracy of our solution, we can perform a verification step. This involves substituting x = 1 back into the original equation and checking if both sides are equal. Let's perform this check:

Original Equation: -5(3 - 7x) = 9x + 11

Substitute x = 1: -5(3 - 7(1)) = 9(1) + 11

Simplify: -5(3 - 7) = 9 + 11

Further Simplify: -5(-4) = 20

Final Check: 20 = 20

As the left side equals the right side, our solution x = 1 is indeed correct. This verification step is a valuable practice in mathematics, providing confidence in the accuracy of our work. It also reinforces the understanding of what a solution to an equation truly means.

In conclusion, the solution to the linear equation -5(3-7x) = 9x + 11 is x = 1, corresponding to option A. 1. This answer represents the value that satisfies the equation, and our detailed step-by-step solution provides a clear understanding of the algebraic process involved.