Solve For $b$.$\frac{b+4}{b+6}=\frac{b+1}{b+2}$There May Be 1 Or 2 Solutions. $b=\square$ Or $b=\square$

Feb 25, 2025 by ADMIN 108 views

Solve for b: A Step-by-Step Guide to Solving the Equation

Introduction

Solving equations is a fundamental concept in mathematics, and it's essential to understand the various techniques involved in solving them. In this article, we will focus on solving a specific equation involving fractions, and we will use a step-by-step approach to find the solutions. The equation we will be solving is:

\frac{b+4}{b+6}=\frac{b+1}{b+2}

Understanding the Equation

Before we start solving the equation, let's understand what it means. The equation states that the ratio of two quantities, $b+4$ and $b+6$ , is equal to the ratio of two other quantities, $b+1$ and $b+2$ . Our goal is to find the value of $b$ that satisfies this equation.

Step 1: Cross-Multiplication

To solve this equation, we will use the technique of cross-multiplication. This involves multiplying both sides of the equation by the denominators of the fractions, which are $b+6$ and $b+2$ . This will eliminate the fractions and give us a simpler equation to work with.

\frac{b+4}{b+6}=\frac{b+1}{b+2}

\Rightarrow (b+4)(b+2) = (b+1)(b+6)

Step 2: Expanding the Equation

Now that we have eliminated the fractions, we can expand the equation by multiplying out the terms on both sides.

(b+4)(b+2) = b^2 + 2b + 4b + 8

(b+1)(b+6) = b^2 + 6b + b + 6

Step 3: Simplifying the Equation

We can simplify the equation by combining like terms on both sides.

b^2 + 4b + 2b + 8 = b^2 + 7b + b + 6

b^2 + 6b + 8 = b^2 + 8b + 6

Step 4: Subtracting b^2 from Both Sides

To get rid of the $b^2$ term on both sides, we can subtract $b^2$ from both sides of the equation.

6b + 8 = 8b + 6

Step 5: Subtracting 6b from Both Sides

Next, we can subtract $6b$ from both sides of the equation to isolate the term with $b$ .

8 = 2b + 6

Step 6: Subtracting 6 from Both Sides

We can simplify the equation further by subtracting 6 from both sides.

2 = 2b

Step 7: Dividing Both Sides by 2

Finally, we can divide both sides of the equation by 2 to solve for $b$ .

1 = b

Conclusion

In this article, we have solved the equation $\frac{b+4}{b+6}=\frac{b+1}{b+2}$ using the technique of cross-multiplication and simplifying the resulting equation. We have found that the solution to the equation is $b=1$ . This means that the value of $b$ that satisfies the equation is 1.

Discussion

The equation we solved in this article is a simple example of a rational equation. Rational equations involve fractions and can be solved using various techniques, including cross-multiplication and simplifying the resulting equation. In this case, we used the technique of cross-multiplication to eliminate the fractions and then simplified the resulting equation to find the solution.

Applications

The technique of solving rational equations is used in a variety of applications, including algebra, calculus, and physics. For example, in algebra, rational equations are used to solve systems of equations and to find the solutions to quadratic equations. In calculus, rational equations are used to find the derivatives of functions and to solve optimization problems. In physics, rational equations are used to model the motion of objects and to solve problems involving forces and energies.

Final Thoughts

In conclusion, solving rational equations is an essential skill in mathematics, and it's used in a variety of applications. In this article, we have solved the equation $\frac{b+4}{b+6}=\frac{b+1}{b+2}$ using the technique of cross-multiplication and simplifying the resulting equation. We have found that the solution to the equation is $b=1$ . This means that the value of $b$ that satisfies the equation is 1.