Solving Equations: A Step-by-Step Guide

Hey there, math enthusiasts! Let's dive into the world of equations, specifically how to solve them step-by-step. We'll be looking at Marlena's work on the equation 2x + 5 = -10 - x and break down each move she made. This is a great way to understand the process and build your confidence in tackling these types of problems. Get ready to flex those math muscles!

Understanding the Basics of Equation Solving

Before we jump into Marlena's solution, let's quickly recap what an equation actually is. Think of it as a balanced scale. On one side, you have an expression, and on the other, you have another expression. The equals sign (=) tells us that both sides are perfectly balanced, or equal in value. Our goal when solving an equation is to find the value of the unknown variable (usually represented by x) that makes the equation true. To do this, we perform operations on both sides of the equation, always keeping it balanced, until we isolate the variable.

The core principle is the golden rule of equation solving: What you do to one side, you MUST do to the other. This ensures that the equation remains balanced throughout the entire process. This means that if you add, subtract, multiply, or divide on one side, you have to do the exact same thing on the other side. Let's make sure we understand the types of operations we will be using in this example. There are many arithmetic operations that you can use to solve an equation. These operations include addition, subtraction, multiplication, and division. When you add or subtract the same number or expression to both sides of an equation, the equation remains equivalent. Likewise, multiplying or dividing both sides of an equation by the same non-zero number maintains the equation's balance. Keeping in mind the golden rule of equation solving, we are now ready to delve into Marlena's steps.

Marlena's Equation: A Detailed Analysis

Let's break down Marlena's approach to solving the equation 2x + 5 = -10 - x. Each step is crucial, and we'll see how she applied the rules to arrive at the solution. We will use the drop-down menus to justify Marlena's work in each step of the process. In this analysis, we will explain each step with details, making it easy to understand the mathematical operations that are used.

Step 1: 2x + 5 = -10 - x becomes 3x + 5 = -10

In the first step, Marlena transformed the original equation 2x + 5 = -10 - x into 3x + 5 = -10. What did she do here? Looking closely, she added x to both sides of the equation. Why? Because the aim is to get all the x terms on one side of the equation. On the left side, we have 2x, and on the right side, we have -x. To eliminate the -x on the right side, Marlena added x to both sides. So, the equation looks like this:

2x + 5 + x = -10 - x + x

When we simplify, -x + x cancels out, leaving us with *

2x + x + 5 = -10*

Then, we can simplify 2x + x as 3x, making the final equation in this step become 3x + 5 = -10. By adding x to both sides, she effectively combined the x terms. This process is crucial in the initial stages of solving the equation to gather all the variable terms on one side. Remember, the golden rule of equations states that you must always perform the same operation on both sides to keep the equation balanced.

Step 2: 3x + 5 = -10 becomes 3x = -15

In the second step, Marlena went from 3x + 5 = -10 to 3x = -15. Here, Marlena wanted to isolate the term with x. This means getting the x term by itself on one side of the equation. To do that, she needed to get rid of the + 5 on the left side. What's the opposite of adding 5? Subtracting 5! So, she subtracted 5 from both sides of the equation. It would look like this:

3x + 5 - 5 = -10 - 5

On the left side, + 5 - 5 cancels out, leaving just 3x. On the right side, -10 - 5 equals -15. So, the simplified equation becomes 3x = -15. Again, this is all about keeping the equation balanced. By subtracting 5 from both sides, Marlena isolated the x term, bringing us closer to finding the value of x.

Step 3: 3x = -15 becomes x = -5

Finally, in the third step, Marlena transformed 3x = -15 into x = -5. This is where she solves for x. The equation now has 3x, which means 3 multiplied by x. To isolate x, you need to do the opposite of multiplication, which is division. So, Marlena divided both sides of the equation by 3. This is what it would look like:

(3x) / 3 = -15 / 3

On the left side, the 3s cancel out, leaving just x. On the right side, -15 / 3 equals -5. Therefore, the final solution is x = -5. Marlena successfully isolated x and found its value. Now she has solved the equation and we know the value of x.

Conclusion: Mastering Equation Solving

There you have it! We've walked through Marlena's solution step-by-step, understanding the reasoning behind each move. Solving equations is like a puzzle, and with practice, you can become a master. Remember the key principles: balance and inverse operations. Always do the same thing to both sides of the equation, and use the opposite operation to undo the operation you want to get rid of. Keep practicing, and you'll find yourself solving equations with confidence. Go ahead, try some equations on your own, and you'll become a pro in no time! Remember, the more you practice, the easier it gets. So, keep at it, guys!