Solving Linear Equations A Step By Step Guide For -12 + 3b - 1 = -5 - B
Decoding Linear Equations
In the realm of mathematics, linear equations stand as fundamental tools for modeling and solving real-world problems. These equations, characterized by their variables raised to the power of one, form the bedrock of various scientific and engineering disciplines. Understanding how to solve linear equations is crucial for anyone seeking to navigate mathematical landscapes. In this article, we will delve into the intricacies of solving a specific linear equation, -12 + 3b - 1 = -5 - b, providing a comprehensive step-by-step guide to unravel its solution.
Before we embark on the solution journey, let's first grasp the essence of linear equations. A linear equation, at its core, is a mathematical statement that asserts the equality between two expressions. These expressions involve variables, which are symbols representing unknown quantities, and constants, which are fixed numerical values. The beauty of linear equations lies in their ability to represent a wide array of scenarios, from calculating the trajectory of a projectile to determining the optimal pricing strategy for a product.
To effectively solve linear equations, we must adhere to a set of fundamental principles. These principles, like the cornerstones of a mathematical edifice, guide us towards the solution. The first principle, often referred to as the addition property of equality, states that adding the same value to both sides of an equation preserves the equality. Similarly, the subtraction property of equality allows us to subtract the same value from both sides without disrupting the balance. The multiplication property of equality extends this concept to multiplication, while the division property of equality caters to division. By judiciously applying these principles, we can manipulate linear equations, isolating the variable and ultimately unveiling its value. The goal is to isolate the variable on one side of the equation, thereby revealing its numerical value. This isolation process involves strategically applying the properties of equality to eliminate terms and simplify the equation. Like a detective piecing together clues, we unravel the equation's solution through a series of logical steps.
Solving the Equation: A Step-by-Step Approach
Now, let's turn our attention to the equation at hand: -12 + 3b - 1 = -5 - b. Our mission is to determine the value of the variable 'b' that satisfies this equation. To accomplish this, we'll embark on a step-by-step journey, employing the principles of equality to isolate 'b'.
Step 1: Simplify Both Sides
The first step in our quest is to simplify both sides of the equation. This involves combining like terms, which are terms that share the same variable or are constants. On the left-hand side, we have two constants, -12 and -1, which can be combined to yield -13. This transforms the left-hand side into 3b - 13. On the right-hand side, we have a constant, -5, and a term involving 'b', -b. Since these terms are unlike, they cannot be combined. Thus, the right-hand side remains -5 - b. Our equation now takes the form:
3b - 13 = -5 - b
Step 2: Isolate the 'b' Terms
Our next objective is to gather all the terms containing 'b' onto one side of the equation. To achieve this, we'll employ the addition property of equality. We'll add 'b' to both sides of the equation. This maneuver eliminates the '-b' term on the right-hand side, effectively consolidating the 'b' terms on the left. Adding 'b' to both sides, we get:
3b - 13 + b = -5 - b + b
Simplifying, we obtain:
4b - 13 = -5
Step 3: Isolate the Constant Terms
Now, our focus shifts to isolating the constant terms on the other side of the equation. To achieve this, we'll again employ the addition property of equality. We'll add 13 to both sides of the equation. This action eliminates the '-13' term on the left-hand side, leaving us with only the term containing 'b'. Adding 13 to both sides, we get:
4b - 13 + 13 = -5 + 13
Simplifying, we obtain:
4b = 8
Step 4: Solve for 'b'
We've reached the final step in our journey: solving for 'b'. To accomplish this, we'll employ the division property of equality. We'll divide both sides of the equation by 4, the coefficient of 'b'. This action isolates 'b', revealing its value. Dividing both sides by 4, we get:
4b / 4 = 8 / 4
Simplifying, we arrive at our solution:
b = 2
Verifying the Solution
To ensure the accuracy of our solution, it's prudent to verify it. This involves substituting the value we obtained for 'b', which is 2, back into the original equation. If the equation holds true, then our solution is indeed correct.
Substituting b = 2 into the original equation, we get:
-12 + 3(2) - 1 = -5 - 2
Simplifying the left-hand side, we have:
-12 + 6 - 1 = -7
Simplifying the right-hand side, we have:
-5 - 2 = -7
Since both sides of the equation evaluate to -7, our solution, b = 2, is verified. We have successfully navigated the equation and unearthed its solution!
Conclusion: Mastering Linear Equations
In this article, we've embarked on a journey to solve the linear equation -12 + 3b - 1 = -5 - b. Through a series of carefully orchestrated steps, we've unveiled the solution, b = 2. This journey has not only provided us with the answer to a specific equation but has also illuminated the fundamental principles that govern the solution of linear equations.
Linear equations, as we've seen, are not mere mathematical abstractions; they are powerful tools that enable us to model and solve real-world problems. By mastering the art of solving linear equations, we equip ourselves with the ability to tackle a wide range of challenges, from determining the optimal dosage of a medication to predicting the growth of a population.
Remember, the key to success in solving linear equations lies in understanding and applying the principles of equality. These principles, like the gears of a well-oiled machine, allow us to manipulate equations, isolating the variable and revealing its value. So, embrace the challenge, hone your skills, and unlock the power of linear equations.
As you continue your mathematical journey, remember that practice makes perfect. The more you engage with linear equations, the more adept you'll become at solving them. So, seek out opportunities to practice, and don't be afraid to ask for help when needed. With dedication and perseverance, you'll master the art of solving linear equations and unlock their vast potential.
Key takeaways:
- Linear equations are mathematical statements that assert the equality between two expressions.
- The principles of equality, including the addition, subtraction, multiplication, and division properties, are essential for solving linear equations.
- Solving a linear equation involves isolating the variable on one side of the equation.
- Verifying the solution by substituting it back into the original equation is crucial.
- Mastering linear equations is a fundamental skill for anyone pursuing mathematics, science, or engineering.
- Practice and perseverance are key to success in solving linear equations.
By embracing these takeaways, you'll be well-equipped to tackle any linear equation that comes your way. So, embark on your mathematical adventure with confidence, and let the power of linear equations guide you!
