Unveiling The Secrets: Solving Equations For 'x'

Hey there, math enthusiasts! Ever felt like equations are secret codes waiting to be cracked? Well, you're in the right place! Today, we're diving deep into the fascinating world of algebra, specifically focusing on how to solve for 'x' in various equations. Don't worry, we'll break it down step by step, making it super easy to understand. We will identify the methods used to solve for 'x' in the following equations: $4x = 20$, $x - 11 = 9$, $5x + 6x = 22$, and $5(x - 2) = 30$. Get ready to become equation-solving pros! This guide is designed to not only provide the solutions but also to build a solid understanding of the methods, ensuring you can tackle similar problems with confidence. We'll explore the underlying principles and common strategies, equipping you with the essential skills to master algebraic equations. By the end, you'll be well-versed in the techniques required to isolate the variable 'x' and find its value. So, let's unlock the secrets of solving for 'x' together!

Decoding the First Equation: $4x = 20$

Let's kick things off with our first equation: $4x = 20$. Our primary goal here is to isolate 'x' on one side of the equation. Right now, 'x' is being multiplied by 4. To undo this, we need to perform the opposite operation, which is division. We'll divide both sides of the equation by 4. The key concept here is maintaining balance. Whatever operation you perform on one side of the equation, you must perform on the other side to keep it equal. The equation, after we divide both sides by 4, becomes: $(4x) / 4 = 20 / 4$. On the left side, the 4s cancel out, leaving us with just 'x'. On the right side, 20 divided by 4 equals 5. Therefore, the solution to this equation is $x = 5$. The method employed here is the division property of equality: dividing both sides of the equation by the same non-zero number maintains the equality. This is a fundamental concept in algebra and is used extensively. It's like having a balanced scale; if you remove a certain weight from one side, you have to remove the same weight from the other side to keep the scale balanced. This property is essential for isolating variables and solving equations.

Now, let's illustrate this with an example. Suppose we have the equation $3x = 12$. To solve for 'x', we would divide both sides by 3: $(3x) / 3 = 12 / 3$. This simplifies to $x = 4$. Another example is $7x = 35$. Here, we divide both sides by 7: $(7x) / 7 = 35 / 7$, which simplifies to $x = 5$. Notice how the division property helps us get 'x' by itself, leading us to the final answer. Understanding this principle is crucial for progressing through more complex algebraic problems. In essence, the method applied to $4x = 20$ involves isolating 'x' through division, a core technique in algebraic manipulations, and it's something you will find yourself using over and over again.

We started with the equation $4x = 20$. We wanted to find the value of x that makes this equation true. Recognizing that 'x' is being multiplied by 4, we used the division property of equality. This meant dividing both sides of the equation by 4 to isolate 'x'. This is similar to the concept of inverse operations. Multiplication and division are inverse operations, just like addition and subtraction. By applying the inverse operation, we can effectively undo the original operation and isolate the variable. This approach is very common and applicable in many equations. Always try to identify the operation affecting the variable and apply the corresponding inverse operation to solve for it. The division property is fundamental because it provides a reliable means to solve linear equations, where the variable is raised to the power of 1. Equations of this kind form the foundation of more complex algebraic concepts.

Solving the Second Equation: $x - 11 = 9$

Let's move on to the second equation: $x - 11 = 9$. In this equation, we see that 11 is being subtracted from 'x'. To isolate 'x', we must perform the inverse operation of subtraction, which is addition. We will add 11 to both sides of the equation. Remember, our golden rule: whatever you do to one side, you must do to the other! So, the equation becomes: $x - 11 + 11 = 9 + 11$. The -11 and +11 on the left side cancel each other out, leaving us with just 'x'. On the right side, 9 + 11 equals 20. Therefore, the solution to this equation is $x = 20$. The method used here is the addition property of equality: adding the same number to both sides of the equation maintains the equality. It is just like the division property of equality.

An example of this property in use is the equation $x - 5 = 10$. To solve this, we add 5 to both sides: $x - 5 + 5 = 10 + 5$, which simplifies to $x = 15$. Here is another one, if $x - 3 = 7$, we add 3 to both sides: $x - 3 + 3 = 7 + 3$, which gives us $x = 10$. As we did before, by performing the inverse operation, we manage to isolate the variable and get closer to finding its value. The addition property of equality is a cornerstone in algebra, offering a simple yet powerful way to solve equations where subtraction is involved. This is because addition and subtraction are inverse operations. The goal is always to manipulate the equation to get 'x' all by itself on one side, and the addition property is a key tool in this process. By adding the same value to both sides, we ensure that the equality remains intact, enabling us to systematically isolate and solve for 'x'.

In our equation, we had $x - 11 = 9$. We used the addition property of equality by adding 11 to both sides. Addition and subtraction are inverse operations, meaning they undo each other. In essence, the addition property helps us to