Solving For 'b': B - 7/4 = -2/3
Let's dive into solving this equation step by step. Our goal is to isolate 'b' on one side of the equation to find its value. This involves basic algebraic principles that are fundamental to mathematics.
Understanding the Equation
The equation we're tackling is: b - 7/4 = -2/3. Here, 'b' is our unknown variable, and we need to figure out what value of 'b' makes this equation true. The numbers 7/4 and -2/3 are constants. To solve for 'b', we'll need to get rid of that - 7/4 on the left side.
Isolating 'b'
To isolate 'b', we want to move the 7/4 to the right side of the equation. Remember the golden rule of algebra: whatever you do to one side of the equation, you must do to the other side. In this case, we're going to add 7/4 to both sides. This will cancel out the - 7/4 on the left, leaving 'b' alone.
So, our equation becomes:
b - 7/4 + 7/4 = -2/3 + 7/4
Simplifying the left side, we get:
b = -2/3 + 7/4
Adding the Fractions
Now, we need to add -2/3 and 7/4. To add fractions, they need to have a common denominator. The least common multiple of 3 and 4 is 12. So, we'll convert both fractions to have a denominator of 12.
To convert -2/3 to a fraction with a denominator of 12, we multiply both the numerator and the denominator by 4:
-2/3 * (4/4) = -8/12
Similarly, to convert 7/4 to a fraction with a denominator of 12, we multiply both the numerator and the denominator by 3:
7/4 * (3/3) = 21/12
Now we can rewrite our equation as:
b = -8/12 + 21/12
Calculating the Result
Adding the fractions, we get:
b = (-8 + 21) / 12
b = 13/12
So, the value of 'b' that satisfies the equation is 13/12. This can also be expressed as a mixed number, which is 1 and 1/12.
Verification
To ensure our solution is correct, we can substitute b = 13/12 back into the original equation:
13/12 - 7/4 = -2/3
First, convert 7/4 to have a denominator of 12:
7/4 * (3/3) = 21/12
Now, substitute back into the equation:
13/12 - 21/12 = -2/3
(-8)/12 = -2/3
Simplifying -8/12 by dividing both the numerator and denominator by 4, we get:
-2/3 = -2/3
The equation holds true, so our solution b = 13/12 is correct.
Conclusion
We've successfully solved for 'b' in the equation b - 7/4 = -2/3. By isolating 'b' and performing the necessary arithmetic operations, we found that b = 13/12. Remember to always verify your solution to ensure accuracy. Understanding these fundamental algebraic principles is crucial for solving more complex equations in the future. Keep practicing, and you'll become more confident in your math skills!
Additional Tips for Solving Equations
When tackling algebraic equations, remember these key strategies:
- Isolate the Variable: The primary goal is always to get the variable by itself on one side of the equation. This usually involves performing inverse operations (addition/subtraction, multiplication/division) to cancel out terms. Always apply these operations to both sides of the equation to maintain balance.
- Simplify Expressions: Before isolating the variable, simplify both sides of the equation as much as possible. Combine like terms, distribute where necessary, and reduce fractions. This makes the equation easier to work with and reduces the chances of making errors.
- Work with Fractions: When dealing with fractions, find a common denominator before adding or subtracting them. This ensures you're working with comparable units. If you prefer, you can also eliminate fractions by multiplying both sides of the equation by the least common multiple of the denominators.
- Check Your Work: After you've found a solution, plug it back into the original equation to verify that it works. This is a crucial step in preventing errors. If the equation doesn't hold true, you know you've made a mistake somewhere and need to re-examine your steps.
- Stay Organized: Keep your work neat and organized, especially when dealing with multi-step equations. Write each step clearly and label your operations. This makes it easier to follow your work and spot any mistakes.
Advanced Techniques
As you progress in algebra, you'll encounter more complex equations that require advanced techniques. Here are a few to keep in mind:
- Factoring: Factoring is a technique used to simplify polynomial expressions and solve quadratic equations. It involves breaking down an expression into its factors, which are expressions that multiply together to give the original expression.
- Quadratic Formula: The quadratic formula is a powerful tool for solving quadratic equations of the form
ax^2 + bx + c = 0. It provides a direct solution for 'x' in terms of the coefficients 'a', 'b', and 'c'. - Systems of Equations: A system of equations is a set of two or more equations with the same variables. Solving a system of equations involves finding values for the variables that satisfy all equations simultaneously.
- Inequalities: Inequalities are mathematical statements that compare two expressions using symbols like < (less than), > (greater than), ≤ (less than or equal to), and ≥ (greater than or equal to). Solving inequalities involves finding the range of values for the variable that make the inequality true.
By mastering these techniques and continuously practicing, you can build a strong foundation in algebra and tackle increasingly challenging problems. Remember that mathematics is a journey, and with each problem you solve, you're expanding your understanding and skills.
In summary, remember the importance of understanding the principles behind each step. Whether it's isolating a variable, finding a common denominator, or using the quadratic formula, a strong grasp of the underlying concepts will empower you to solve a wide range of problems. And remember, practice makes perfect. The more you work with equations, the more comfortable and confident you'll become.
Finally, explore additional resources and seek help when needed. There are numerous online tutorials, textbooks, and educators available to support your learning journey. Don't hesitate to ask questions and collaborate with others. Khan Academy Algebra is an excellent place to start. Happy solving!
