Solving 7(x - 2) = 4(x - 5): A Step-by-Step Guide

Alright guys, let's break down this math problem together! We're going to solve the equation 7(x - 2) = 4(x - 5) step-by-step. Don't worry, it's easier than it looks. We'll go through each stage slowly and clearly, so you can follow along and understand exactly how to get to the answer.

1. Understanding the Equation

First, let's understand what the equation 7(x - 2) = 4(x - 5) actually means. In algebra, an equation is a statement that two expressions are equal. Here, we have two expressions: 7(x - 2) on the left side and 4(x - 5) on the right side. Our goal is to find the value of 'x' that makes these two expressions equal. This involves several algebraic manipulations to isolate 'x' on one side of the equation.

The equation involves the distributive property, which is a fundamental concept in algebra. The distributive property states that a(b + c) = ab + ac. This property will be crucial in simplifying both sides of our equation. We'll be multiplying the numbers outside the parentheses (7 and 4) by each term inside the parentheses (x and -2 on the left, and x and -5 on the right). This step is essential to remove the parentheses and combine like terms later on.

Additionally, we need to understand the concept of inverse operations. To isolate 'x', we'll use inverse operations to undo the operations that are being applied to 'x'. For example, if 'x' is being added to a number, we'll subtract that number from both sides of the equation. If 'x' is being multiplied by a number, we'll divide both sides by that number. This ensures that the equation remains balanced and that we're accurately finding the value of 'x'. Remember, whatever operation you perform on one side of the equation, you must also perform on the other side to maintain equality. This principle is the cornerstone of solving algebraic equations.

2. Applying the Distributive Property

The distributive property is our best friend here! Let's apply it to both sides of the equation:

• Left side: 7(x - 2) = 7 * x - 7 * 2 = 7x - 14
• Right side: 4(x - 5) = 4 * x - 4 * 5 = 4x - 20

So, our equation now looks like this: 7x - 14 = 4x - 20. Applying the distributive property correctly is super important. A small mistake here can throw off the entire solution. Double-check that you've multiplied the number outside the parentheses by each term inside. It’s a common error to forget to distribute to the second term, especially when it involves a negative sign. So, keep a close eye on those signs!

After applying the distributive property, you should have a clearer picture of the equation. The parentheses are gone, and now you have a linear equation with 'x' terms and constant terms on both sides. This sets the stage for the next step, which involves collecting like terms. By simplifying the equation in this way, you make it easier to isolate 'x' and find its value. The distributive property is truly the key to unlocking this type of equation, so make sure you're comfortable with it before moving on.

Understanding and correctly applying the distributive property is not just crucial for this particular problem but for a wide range of algebraic problems. It forms the foundation for simplifying expressions, solving equations, and manipulating formulas. Make sure to practice with various examples to solidify your understanding. Once you master this property, you'll find that many algebraic problems become much more manageable. Remember, algebra is all about breaking down complex problems into simpler steps, and the distributive property is a prime example of this approach.

3. Moving the 'x' Terms

Our goal is to get all the 'x' terms on one side of the equation. Let's subtract 4x from both sides:

7x - 14 - 4x = 4x - 20 - 4x

This simplifies to: 3x - 14 = -20

Moving the 'x' terms is a strategic step that brings us closer to isolating 'x'. By subtracting 4x from both sides, we eliminate the 'x' term from the right side of the equation, consolidating all 'x' terms on the left side. This step is based on the principle of maintaining balance in the equation; whatever we do to one side, we must also do to the other. This ensures that the equality remains true throughout the solving process. After this step, the equation becomes simpler and easier to manipulate.

It's important to choose the right term to move to make the process as efficient as possible. In this case, subtracting 4x from both sides is the most straightforward way to eliminate the 'x' term from the right side. Alternatively, we could have subtracted 7x from both sides, but this would result in a negative coefficient for 'x', which might be slightly more complicated to deal with later on. By making a smart choice at this stage, we can streamline the rest of the solution.

Furthermore, remember to combine like terms after moving the 'x' terms. In our example, we had to combine 7x and -4x to get 3x. This step is crucial for simplifying the equation and making it easier to work with. Always double-check your calculations to ensure that you're combining the terms correctly. A small mistake in this step can lead to an incorrect final answer. So, pay close attention to the signs and coefficients when combining like terms.

4. Isolating 'x'

Now, let's isolate 'x' by getting rid of the -14. Add 14 to both sides:

3x - 14 + 14 = -20 + 14

This gives us: 3x = -6

Isolating 'x' involves removing any constants that are added to or subtracted from the 'x' term. In our case, we have -14 on the left side of the equation. To eliminate it, we perform the inverse operation, which is adding 14 to both sides. This step is based on the same principle of maintaining balance that we've been using throughout the solving process. By adding 14 to both sides, we ensure that the equation remains equal and that we're accurately isolating 'x'. After this step, the equation becomes even simpler, with only the 'x' term and a constant term remaining.

It's important to remember that you're always trying to