Solving Equations: A Step-by-Step Guide

Hey everyone! Today, we're diving into a math problem that might seem a bit intimidating at first glance, but trust me, we'll break it down into manageable chunks. The equation we're tackling is: $-(5-(a+1))=9-(5-(2 a-3))$. Our goal? To find the value of 'a' that makes this equation true. Let's get started, shall we?

Understanding the Basics: Equations and Variables

Before we jump into the nitty-gritty, let's make sure we're all on the same page. An equation is simply a mathematical statement that shows two expressions are equal. Think of it like a seesaw; both sides must balance. Our equation has an equals sign (=), indicating the expressions on both sides are equivalent. The expressions involve variables, which are letters (in this case, 'a') representing unknown values. Our mission is to isolate the variable, 'a', on one side of the equation to discover its value.

Now, why is solving equations important, you ask? Well, it's a fundamental skill in mathematics. Whether you're balancing a checkbook, calculating the best deal at the grocery store, or even understanding complex scientific principles, equations are at the heart of it all. They allow us to model and solve real-world problems. They're also the building blocks for more advanced mathematical concepts like algebra, calculus, and beyond. So, mastering this skill is like building a solid foundation for all your future math endeavors! Furthermore, they are applicable to many fields. For example, in computer science and data science, equations are used to create complex algorithms and models. In engineering, they are used to design systems and infrastructure. Even in fields like economics and finance, equations are used to model market behaviors and financial instruments. Solving equations equips us with the tools necessary to analyze and interpret information, identify patterns, and make informed decisions, which proves to be very useful in many aspects of life.

Step-by-Step Solution: Unraveling the Equation

Alright, let's get down to business and solve this equation. We'll break it down into small steps to make it easier to follow. Remember the order of operations? PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction) is our guiding star here. It tells us the order in which we should simplify expressions. Our goal is to isolate 'a' on one side of the equation. Here’s how we'll do it:

Step 1: Simplify Parentheses

First, we tackle the parentheses. Let’s start with the left side of the equation: $-(5-(a+1))$. Inside the inner parentheses, we have $(a+1)$. Nothing to simplify there yet, so we'll leave it as is. Now let's simplify the expression $- (5 - (a + 1))$. The negative sign in front means we'll be distributing it to the terms inside the parentheses. So we have: $-(5 - a - 1)$. Continuing this, we get: $-5 + a + 1$. This simplifies to $a - 4$. Now let's move over to the right side of the equation: $9 - (5 - (2a - 3))$. First, we simplify the inner parenthesis $(2a - 3)$. Then, we simplify the expression $-(5 - (2a - 3))$. Distribute the negative sign to get $-(5 - 2a + 3)$. This gives us $-5 + 2a - 3$. This simplifies to $2a - 8$. Now our equation looks like this:

$a - 4 = 2a - 8$

Step 2: Combine Like Terms

Now we're going to combine like terms. Our objective is to get all the 'a' terms on one side of the equation and the constant numbers on the other side. This helps us isolate 'a'. We have 'a' on both sides of the equation. Let's subtract 'a' from both sides of the equation. This gives us:

$a - a - 4 = 2a - a - 8$

Simplifying this, we get:

$-4 = a - 8$

Step 3: Isolate the Variable

Our next move is to isolate 'a'. Currently, we have '-8' being subtracted from 'a'. To get 'a' alone, we'll do the opposite operation: add 8 to both sides of the equation. Remember, whatever we do to one side, we must do to the other to keep the equation balanced. So, we'll add 8 to both sides:

$-4 + 8 = a - 8 + 8$

This simplifies to:

$4 = a$

Which can also be written as $a = 4$

Step 4: Verification (Checking Your Answer)

We've found our solution, but we should always double-check our work. This is the verification step and is a super important practice! We'll substitute 'a = 4' back into the original equation to see if it holds true. Remember, the original equation was: $-(5-(a+1))=9-(5-(2 a-3))$.

Let’s substitute 'a' with 4 on the left side: $-(5 - (4 + 1))$. This simplifies to $-(5 - 5)$, which equals $-(0)$, which is equal to 0.

Now, let's work on the right side of the equation. We have: $9 - (5 - (2 * 4 - 3))$. This simplifies to $9 - (5 - (8 - 3))$, which equals $9 - (5 - 5)$, and this simplifies to $9 - 0$, which is equal to 9.

Oops! We made a mistake. Let's re-evaluate our solution. $-(5-(a+1))=9-(5-(2 a-3))$.

Let's go back and carefully analyze our steps and pinpoint our error. We've simplified the equation to $a-4=2a-8$, and then we subtract a from both sides, which gives us $-4 = a - 8$. We need to isolate 'a'. This time, let's add 8 to both sides to solve for 'a'. This gives us:

$a-8+8 = -4+8$

This simplifies to

$a = 4$

Let's re-verify our answer.

Left side: $-(5-(4+1)) = -(5-5) = 0$

Right side: $9-(5-(2*4-3)) = 9 - (5 - 5) = 9 - 0 = 9$

So it seems like there was an error in our work earlier. Let's go through the steps again.

Our equation, after simplifying, was $a - 4 = 2a - 8$. The next step should be moving all the variables to one side of the equation. We subtract 'a' on both sides.

$a-4-a = 2a-8-a$

$-4 = a - 8$

Now, add 8 to both sides of the equation to isolate 'a'.

$-4 + 8 = a - 8 + 8$

This becomes

$4 = a$

Let's verify again.

Left Side: $-(5-(4+1)) = 0$

Right Side: $9-(5-(2*4-3)) = 9-0 = 9$

We did it again! Let's carefully re-evaluate our steps.

$-(5-(a+1)) = 9-(5-(2a-3))$

$-5 + a + 1 = 9 - (5 - 2a + 3)$

$a - 4 = 9 - 8 + 2a$

$a - 4 = 1 + 2a$

$-4-1=2a-a$

$-5 = a$

Therefore, a = -5

Let's verify this:

Left Side: $-(5 - (-5 + 1)) = -(5 - (-4)) = -(5 + 4) = -9$

Right Side: $9 - (5 - (2*(-5) - 3)) = 9 - (5 - (-10 - 3)) = 9 - (5 - (-13)) = 9 - 18 = -9$

So, our solution is $a = -5$

Conclusion: Practice Makes Perfect!

Congratulations, we did it! The solution to the equation $-(5-(a+1))=9-(5-(2 a-3))$ is $a = -5$. We've learned the process of solving this equation and hopefully gained some valuable insights. The key takeaways from this exercise include the importance of carefully following the order of operations, paying attention to the signs, and double-checking your work. Remember, practice is key to mastering any math skill. So, grab some more equations and start solving! The more you practice, the more comfortable and confident you'll become. Keep up the great work, and don't be afraid to ask for help if you need it. Math can be tricky, but with perseverance and the right approach, you can conquer any equation! Keep practicing, and you'll become a pro in no time! We have reviewed the steps to verify the solution by plugging the value of 'a' into the original equation, verifying both sides. We have now solved the equation and verified the solution! Happy calculating!