Solving equations is a fundamental concept in mathematics that allows us to find the unknown value of a variable. It is a process that involves manipulating the given equation using various mathematical operations to isolate the variable and determine its value. In this article, we will focus on solving a specific equation: 3/2b+5 = 17. Understanding the steps involved in solving equations is crucial for developing strong mathematical skills.
The Equation: 3/2b+5 = 17
The equation we will be working with is 3/2b+5 = 17. This equation contains a variable ‘b’ that we need to solve for. By finding the value of ‘b’, we can determine the solution to the equation. However, before we dive into the solution, it is important to understand the different components of the equation.
Understanding the Equation
To solve the equation 3/2b+5 = 17, we need to break it down and understand each component. The equation consists of three parts: 3/2b, +5, and 17.
The term 3/2b represents a fraction multiplied by ‘b’. The fraction 3/2 is multiplied by the variable ‘b’.
The term +5 is a constant value that is added to the fraction 3/2b.
Finally, the term 17 represents the constant value on the other side of the equation.
The Goal: Solving for ‘b’
The ultimate goal of solving the equation 3/2b+5 = 17 is to find the value of ‘b’ that makes the equation true. By isolating the variable ‘b’ on one side of the equation, we can determine its value and solve the equation.
Now that we have a clear understanding of the equation and its components, let’s move on to the step-by-step solution to find the value of ‘b’.
Understanding the Equation
To effectively solve an equation, it is crucial to understand its components and their meanings. In this section, we will break down the equation 3/2b+5 = 17 and explain the significance of each element.
Breaking Down the Equation
Let’s analyze the equation 3/2b+5 = 17 step by step. This equation consists of three main components: 3/2b, +5, and 17.
The term 3/2b represents a fraction multiplied by the variable ‘b’. It is essential to recognize that the fraction 3/2 is multiplied by ‘b’. The addition sign (+) separates this term from the next component.
The term +5 is a constant value added to the previous term. It is crucial to differentiate between the variable ‘b’ and the constant value 5.
Lastly, the number 17 represents the result or the outcome of the equation. The goal of solving this equation is to find the value of ‘b’ that makes the equation true.
The Goal of Solving the Equation
The primary objective of solving an equation is to determine the value of the variable in question. In this case, our goal is to find the value of ‘b’ that satisfies the equation 3/2b+5 = 17.
By solving the equation, we can uncover the specific value of ‘b’ that, when substituted into the equation, makes it a valid statement. This process allows us to find the solution to the equation and gain a deeper understanding of the relationship between the variables involved.
Importance of Understanding the Equation
Understanding the components and the goal of solving an equation is vital for several reasons. Firstly, it enables us to approach the problem with clarity and confidence. By comprehending the equation’s structure, we can apply the appropriate techniques and steps to arrive at the solution accurately.
Furthermore, understanding the equation allows us to identify any potential challenges or obstacles that may arise during the solving process. It helps us anticipate the complexities that may arise and devise strategies to overcome them.
Lastly, comprehending the equation fosters a deeper understanding of mathematical concepts and principles. By engaging with equations and their solutions, we enhance our problem-solving skills and develop a more profound appreciation for the intricacies of mathematics.
In conclusion, understanding the equation 3/2b+5 = 17 involves breaking down its components, recognizing their meanings, and grasping the goal of solving the equation. This understanding is crucial for accurately solving the equation and gaining a deeper comprehension of mathematical concepts. By emphasizing the significance of understanding equations, we encourage readers to practice solving similar equations to enhance their skills and confidence in mathematics.
Step-by-Step Solution
Solving equations may seem daunting at first, but with a step-by-step approach, it becomes much more manageable. Let’s dive into the process of solving the equation 3/2b+5 = 17 and find the value of ‘b’.
Start with simplifying the equation
To begin, we need to simplify the equation by eliminating the fraction. In this case, we have the term 3/2b, which can be simplified by multiplying both sides of the equation by the denominator, 2.
2 * (3/2b+5) = 2 * 17
By doing this, the denominator cancels out, and we are left with:
3b + 10 = 34
Combine like terms and isolate the variable ‘b’
Next, we combine the like terms on the left side of the equation. In this case, we have 3b and 10. To isolate the variable ‘b’, we subtract 10 from both sides of the equation.
3b + 10 - 10 = 34 - 10
Simplifying further, we get:
3b = 24
Solve for ‘b’
To solve for ‘b’, we need to divide both sides of the equation by the coefficient of ‘b’, which is 3.
(3b)/3 = 24/3
Simplifying the equation, we find:
b = 8
Therefore, the value of ‘b’ that satisfies the equation 3/2b+5 = 17 is 8.
IV. Checking the Solution
Now that we have found the value of ‘b’, it is essential to check if our solution holds true by substituting it back into the original equation.
3/2(8) + 5 = 17
Simplifying the equation, we get:
12 + 5 = 17
As we can see, the equation holds true with the calculated value of ‘b’. This step is crucial as it ensures the accuracy of our solution.
In conclusion, solving equations requires a systematic approach to arrive at the correct solution. By breaking down the equation, simplifying it, and isolating the variable, we can find the value that satisfies the equation. Checking the solution is equally important to validate our answer.
Understanding the process of solving equations is crucial as it forms the foundation for various mathematical concepts and real-life applications. By practicing similar equations, you can improve your problem-solving skills and gain confidence in tackling more complex equations.
Remember, solving equations is a skill that can be honed with practice. So, keep practicing and embrace the beauty of mathematical problem-solving!
Checking the Solution
After successfully solving the equation 3/2b+5 = 17 and obtaining a value for ‘b’, it is important to check the solution to ensure its accuracy. This step is crucial in mathematics as it helps to validate the solution and confirms that the equation holds true with the calculated value.
Substitute the found value of ‘b’ back into the original equation
To check the solution, we need to substitute the value of ‘b’ that we obtained back into the original equation. In this case, the original equation is 3/2b+5 = 17. Let’s assume that the value of ‘b’ we found is 4.
Substituting ‘b’ with 4 in the equation, we get:
3/2(4) + 5 = 17
Verify if the equation holds true with the calculated value
Now, let’s simplify the equation and see if it holds true with the calculated value of ‘b’.
3/2(4) is equal to 3/8, and when we add 5 to it, we get:
3/8 + 5 = 17
To add these fractions, we need to find a common denominator. In this case, the common denominator is 8. So, we rewrite the equation as:
3/8 + 40/8 = 17
Adding the fractions, we get:
43/8 = 17
To check if this equation is true, we can simplify the left side of the equation by dividing the numerator by the denominator:
43 ÷ 8 = 5.375
As we can see, the left side of the equation simplifies to 5.375, which is not equal to 17. Therefore, the solution we obtained is incorrect.
Re-evaluate the solution
If the equation does not hold true with the calculated value, it means that there was an error in our solution. In such cases, we need to re-evaluate our steps and identify where we made a mistake.
In the process of solving equations, it is common to make errors such as calculation mistakes, incorrect distribution of multiplication, or missing a step. It is crucial to carefully review our work and identify any mistakes that may have occurred.
Checking the solution is an essential step in solving equations. It helps to validate the solution and ensure that the equation holds true with the calculated value. By carefully substituting the found value back into the original equation and verifying its accuracy, we can identify any errors and re-evaluate our steps if necessary.
It is important to note that making mistakes while solving equations is common, especially when dealing with complex equations. However, through practice and careful attention to detail, we can improve our problem-solving skills and minimize errors.
In conclusion, understanding the process of solving equations and checking the solution are fundamental in mathematics. By following the step-by-step solution and verifying the solution’s accuracy, we can confidently solve equations and obtain accurate results. So, keep practicing and honing your skills to become a proficient equation solver.
