Solving Linear Equations Step-by-Step With Properties Of Equality

In this article, we will explore the process of solving linear equations, focusing on a step-by-step approach to ensure clarity and accuracy. We'll dissect an example equation, identifying the properties of equality used at each stage. Understanding these properties is crucial for manipulating equations and isolating the variable. Linear equations are fundamental in mathematics, and mastering their solution is essential for success in algebra and beyond.

Understanding the Problem

Let's consider the equation: $-2(5x + 8) = 14 + 6x$

This equation presents a typical scenario where we need to solve for the unknown variable, 'x'. The solution involves simplifying the equation and isolating 'x' on one side. To achieve this, we'll follow a series of steps, each justified by a property of equality. These properties allow us to perform operations on both sides of the equation without changing its solution. The process begins with the distributive property, a cornerstone of algebraic manipulation.

Step 1: Applying the Distributive Property

The first step in solving the equation involves applying the distributive property. This property states that a(b + c) = ab + ac. In our equation, we need to distribute the -2 across the terms inside the parentheses:

$$-2(5x + 8) = -2 * 5x + (-2) * 8 = -10x - 16$$

Therefore, the equation becomes:

$$-10x - 16 = 14 + 6x$$

This step simplifies the equation by removing the parentheses, making it easier to work with. The distributive property is essential for handling expressions involving parentheses and is a fundamental tool in algebraic manipulation. By correctly applying this property, we ensure that the equation remains balanced and that we are progressing towards the correct solution. Now we have a more manageable equation, free of parentheses, setting the stage for the next steps in isolating the variable 'x'.

Step 2: Isolating the Variable Term

Our goal is to isolate the variable term, which in this case is '-10x' on the left side and '6x' on the right side. To do this, we need to eliminate the constant term, -16, from the left side. We can achieve this by using the Addition Property of Equality. This property states that if we add the same value to both sides of an equation, the equation remains balanced. Adding 16 to both sides of the equation gives us:

$$-10x - 16 + 16 = 14 + 6x + 16$$

Simplifying this, we get:

$$-10x = 30 + 6x$$

This step moves us closer to isolating 'x'. By adding 16 to both sides, we've effectively canceled out the -16 on the left side, leaving us with only the term containing 'x'. This is a crucial step in solving for 'x', as it concentrates the variable term on one side of the equation. The addition property of equality is a powerful tool in equation solving, allowing us to manipulate equations while maintaining their balance and ensuring that we arrive at the correct solution.

Step 3: Combining Like Terms

Now we need to get all the 'x' terms on one side of the equation. We can achieve this by subtracting 6x from both sides. This again utilizes the Subtraction Property of Equality, which, like the addition property, allows us to perform the same operation on both sides without changing the equation's solution. Subtracting 6x from both sides gives us:

$$-10x - 6x = 30 + 6x - 6x$$

Simplifying this, we get:

$$-16x = 30$$

This step is crucial for isolating 'x'. By subtracting 6x from both sides, we've eliminated the 'x' term from the right side, leaving us with only a constant. This brings us closer to our goal of having 'x' by itself on one side of the equation. Combining like terms is a fundamental algebraic technique, and the subtraction property of equality is a key tool in this process. By correctly applying this property, we maintain the equation's balance and move closer to the final solution.

Step 4: Isolating the Variable

Finally, to isolate 'x', we need to get rid of the coefficient -16. We can do this by dividing both sides of the equation by -16. This utilizes the Division Property of Equality, which states that dividing both sides of an equation by the same non-zero value maintains the equation's balance. Dividing both sides by -16 gives us:

$$\frac{-16x}{-16} = \frac{30}{-16}$$

Simplifying this, we get:

$$x = -\frac{15}{8}$$

Therefore, the solution to the equation is x = -15/8. This final step completes the process of solving for 'x'. By dividing both sides by the coefficient of 'x', we isolate the variable and determine its value. The division property of equality is a vital tool in equation solving, allowing us to undo multiplication and isolate the variable. This step, combined with the previous steps, demonstrates the systematic approach to solving linear equations.

Properties of Equality

Throughout the solution, we used several properties of equality:

• Distributive Property: a(b + c) = ab + ac
• Addition Property of Equality: If a = b, then a + c = b + c
• Subtraction Property of Equality: If a = b, then a - c = b - c
• Division Property of Equality: If a = b, then a / c = b / c (where c ≠ 0)

Understanding and applying these properties is crucial for solving equations correctly. Each property allows us to manipulate the equation in a way that maintains its balance and leads us closer to the solution. These properties are not just rules to memorize; they are the foundation upon which we build our understanding of equation solving. By grasping the logic behind them, we can confidently tackle a wide range of algebraic problems.

Conclusion

Solving linear equations involves a systematic application of properties of equality. By understanding these properties and applying them carefully, we can isolate the variable and find the solution. In this article, we've walked through the steps of solving a specific equation, highlighting the use of the distributive property, the addition and subtraction properties of equality, and the division property of equality. Mastering these techniques is essential for success in algebra and beyond. The ability to solve linear equations is a fundamental skill that opens doors to more advanced mathematical concepts and problem-solving scenarios. Practice and a solid understanding of these principles will empower you to confidently tackle any linear equation that comes your way.

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