Solving |1 + 5b| = 13 A Step-by-Step Guide To Finding B Values
In mathematics, solving equations involving absolute values can sometimes seem daunting, but with a clear understanding of the underlying principles, it becomes a manageable task. This article delves into the process of solving the equation |1 + 5b| = 13, providing a step-by-step guide to finding all possible values of b in their simplest form. We'll explore the fundamental concept of absolute value, break down the equation into manageable parts, and demonstrate the algebraic techniques required to arrive at the solutions. This comprehensive guide aims to equip you with the knowledge and skills to confidently tackle similar problems in the future. So, let's embark on this mathematical journey and unlock the solutions together.
Understanding Absolute Value
Before diving into the equation, it's crucial to grasp the concept of absolute value. The absolute value of a number represents its distance from zero on the number line, irrespective of direction. In simpler terms, it's the magnitude of the number without considering its sign. For instance, the absolute value of 5, denoted as |5|, is 5, and the absolute value of -5, denoted as |-5|, is also 5. This fundamental property is key to solving equations involving absolute values.
When we encounter an equation like |1 + 5b| = 13, it signifies that the expression inside the absolute value bars, which is (1 + 5b), is either 13 units away from zero in the positive direction or 13 units away in the negative direction. This gives rise to two distinct possibilities that we need to consider to find all possible values of b. One possibility is that (1 + 5b) equals 13, and the other is that (1 + 5b) equals -13. By addressing both scenarios, we ensure that we capture all potential solutions for b.
Understanding this duality is paramount in solving absolute value equations. It's not simply about finding one solution but about recognizing that the absolute value function introduces a branching path, each leading to a potential answer. The subsequent steps in solving the equation will involve treating these possibilities as separate linear equations, which we will then solve using standard algebraic techniques. By keeping the concept of absolute value as distance from zero in mind, we can approach these problems with clarity and precision.
Breaking Down the Equation
The absolute value equation |1 + 5b| = 13 implies that the expression (1 + 5b) can be either 13 or -13. This is because the absolute value of a number is its distance from zero, and both 13 and -13 are 13 units away from zero. To solve for b, we need to consider both possibilities separately, which will lead to two distinct linear equations.
Case 1: 1 + 5b = 13
In the first case, we assume that the expression inside the absolute value, (1 + 5b), is equal to 13. This gives us a linear equation that we can solve using standard algebraic methods. Our goal is to isolate b on one side of the equation. To do this, we first subtract 1 from both sides of the equation, which gives us 5b = 12. Then, we divide both sides by 5 to solve for b, resulting in b = 12/5. This is one potential solution for b.
Case 2: 1 + 5b = -13
The second case arises from the possibility that the expression (1 + 5b) could be equal to -13. Again, this leads to a linear equation that we can solve for b. Similar to the first case, we begin by subtracting 1 from both sides of the equation, which gives us 5b = -14. Next, we divide both sides by 5 to isolate b, resulting in b = -14/5. This is the second potential solution for b.
By considering both cases, we have accounted for the two possible scenarios dictated by the absolute value. Each case provides a unique value for b that satisfies the original equation. The next step is to express these solutions in their simplest form and verify them to ensure accuracy. Breaking down the equation in this manner allows us to systematically address the absolute value and arrive at the complete set of solutions.
Solving the Two Linear Equations
As established earlier, the absolute value equation |1 + 5b| = 13 leads to two separate linear equations:
- 1 + 5b = 13
- 1 + 5b = -13
We will now proceed to solve each of these equations step-by-step.
Solving 1 + 5b = 13
To isolate b in this equation, we first need to eliminate the constant term on the left side. This is achieved by subtracting 1 from both sides of the equation. This operation maintains the balance of the equation while moving us closer to isolating b. Subtracting 1 from both sides gives us:
1 + 5b - 1 = 13 - 1
This simplifies to:
5b = 12
Now, to completely isolate b, we need to remove the coefficient of 5. We do this by dividing both sides of the equation by 5. This is a fundamental algebraic operation that preserves the equality while isolating the variable of interest. Dividing both sides by 5, we get:
5b / 5 = 12 / 5
This simplifies to:
b = 12/5
Thus, the first solution for b is 12/5. This is a fraction in its simplest form, as 12 and 5 have no common factors other than 1. This solution represents one of the values of b that makes the absolute value equation true.
Solving 1 + 5b = -13
We follow a similar process to solve the second equation. Our goal remains to isolate b on one side of the equation. We begin by subtracting 1 from both sides of the equation to eliminate the constant term on the left side. Subtracting 1 from both sides gives us:
1 + 5b - 1 = -13 - 1
This simplifies to:
5b = -14
Next, we divide both sides of the equation by 5 to remove the coefficient of b and isolate the variable. Dividing both sides by 5, we get:
5b / 5 = -14 / 5
This simplifies to:
b = -14/5
Thus, the second solution for b is -14/5. This is also a fraction in its simplest form, as -14 and 5 have no common factors other than 1. This solution represents the other value of b that satisfies the original absolute value equation.
By solving these two linear equations, we have found the two possible values for b that make the absolute value equation |1 + 5b| = 13 true. These solutions, 12/5 and -14/5, are expressed in their simplest forms and represent the complete solution set for the equation.
Expressing the Solutions in Simplest Form
After solving the two linear equations, we obtained the solutions b = 12/5 and b = -14/5. Now, we need to ensure that these solutions are expressed in their simplest form. A fraction is in its simplest form when the numerator and denominator have no common factors other than 1. In other words, the fraction cannot be further reduced.
Checking 12/5
For the fraction 12/5, we need to determine if 12 and 5 share any common factors. The factors of 12 are 1, 2, 3, 4, 6, and 12. The factors of 5 are 1 and 5. The only common factor between 12 and 5 is 1. Therefore, the fraction 12/5 is already in its simplest form. It cannot be reduced any further, as the numerator and denominator are relatively prime.
Checking -14/5
Similarly, for the fraction -14/5, we need to check if 14 and 5 have any common factors. The factors of 14 are 1, 2, 7, and 14. As we established earlier, the factors of 5 are 1 and 5. The only common factor between 14 and 5 is 1. Consequently, the fraction -14/5 is also in its simplest form. The negative sign does not affect the process of simplifying fractions; we only focus on the numerical values of the numerator and denominator.
Since both solutions, 12/5 and -14/5, are already in their simplest forms, we do not need to perform any further reduction. These fractions represent the final solutions for b in the equation |1 + 5b| = 13. Expressing solutions in their simplest form is a crucial step in mathematical problem-solving, as it ensures clarity and conciseness in the final answer. It also demonstrates a complete understanding of the properties of numbers and fractions.
Verifying the Solutions
To ensure the accuracy of our solutions, it's essential to verify them by substituting them back into the original equation. This process confirms that the values we found for b indeed satisfy the equation |1 + 5b| = 13. We will substitute each solution separately and check if the equation holds true.
Verifying b = 12/5
Let's substitute b = 12/5 into the original equation:
|1 + 5(12/5)| = 13
First, we perform the multiplication inside the absolute value:
|1 + 12| = 13
Then, we add the numbers:
|13| = 13
The absolute value of 13 is indeed 13, so the equation holds true:
13 = 13
This confirms that b = 12/5 is a valid solution for the equation.
Verifying b = -14/5
Now, let's substitute b = -14/5 into the original equation:
|1 + 5(-14/5)| = 13
Again, we start by performing the multiplication inside the absolute value:
|1 - 14| = 13
Then, we subtract the numbers:
|-13| = 13
The absolute value of -13 is 13, so the equation holds true:
13 = 13
This confirms that b = -14/5 is also a valid solution for the equation.
By verifying both solutions, we have demonstrated that they satisfy the original equation |1 + 5b| = 13. This step is crucial in problem-solving as it provides assurance that the solutions are correct and that no errors were made during the algebraic manipulations. Verification reinforces the understanding of the problem and the solution process, making it an integral part of mathematical practice.
Conclusion
In this comprehensive guide, we have successfully solved the absolute value equation |1 + 5b| = 13. We began by understanding the concept of absolute value and its implications for solving equations. We then broke down the equation into two separate linear equations, each representing a possible scenario dictated by the absolute value. We solved these equations using standard algebraic techniques, isolating b in each case. The solutions obtained were b = 12/5 and b = -14/5. We ensured that these solutions were expressed in their simplest form, confirming that no further reduction was possible. Finally, we verified the solutions by substituting them back into the original equation, demonstrating that they indeed satisfy the equation.
Solving absolute value equations requires a clear understanding of the absolute value concept and the ability to handle multiple possibilities. By systematically addressing each possibility and applying algebraic principles, we can arrive at accurate solutions. The process involves breaking down the problem, solving linear equations, simplifying fractions, and verifying the results. These skills are essential in mathematics and are applicable to a wide range of problems.
The solutions to the equation |1 + 5b| = 13 are b = 12/5 and b = -14/5. These values represent the complete set of solutions for b in the given equation. This exercise illustrates the importance of a methodical approach to problem-solving and the significance of verifying solutions to ensure accuracy. The ability to solve absolute value equations is a valuable asset in mathematical studies and beyond.