Solving -2(bx-5)=16 Value Of X In Terms Of B And When B Is 3

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Solve the equation $-2(bx-5)=16$. Find the value of $x$ in terms of $b$, and then find the value of $x$ when $b=3$.

This article provides a detailed, step-by-step solution to the equation $-2(bx - 5) = 16$, focusing on finding the value of $x$ in terms of $b$ and then calculating the specific value of $x$ when $b = 3$. Whether you're a student grappling with algebraic equations or simply looking to refresh your math skills, this guide will help you understand the underlying concepts and techniques involved.

Finding the Value of x in Terms of b

To find the value of $x$ in terms of $b$, we need to isolate $x$ on one side of the equation. The given equation is:

−2(bx−5)=16-2(bx - 5) = 16

Our first step involves distributing the $-2$ across the terms inside the parentheses. This means multiplying both $bx$ and $-5$ by $-2$:

−2∗bx+(−2)∗(−5)=16-2 * bx + (-2) * (-5) = 16

This simplifies to:

−2bx+10=16-2bx + 10 = 16

Now, our goal is to isolate the term containing $x$, which is $-2bx$. To do this, we need to get rid of the $+10$ on the left side of the equation. We can achieve this by subtracting $10$ from both sides of the equation. This maintains the balance of the equation:

−2bx+10−10=16−10-2bx + 10 - 10 = 16 - 10

This simplifies to:

−2bx=6-2bx = 6

Now that we have $-2bx$ isolated, we need to isolate $x$ itself. Currently, $x$ is being multiplied by $-2b$. To undo this multiplication, we perform the inverse operation, which is division. We divide both sides of the equation by $-2b$:

−2bx−2b=6−2b\frac{-2bx}{-2b} = \frac{6}{-2b}

This simplifies to:

x=6−2bx = \frac{6}{-2b}

We can further simplify the fraction by dividing both the numerator and the denominator by their greatest common divisor, which is $-2$:

x=6÷−2−2b÷−2x = \frac{6 \div -2}{-2b \div -2}

This gives us:

x=−3bx = \frac{-3}{b}

Therefore, the value of $x$ in terms of $b$ is:

x=−3bx = -\frac{3}{b}

This expression tells us how the value of $x$ depends on the value of $b$. For any given value of $b$ (except 0, as division by zero is undefined), we can substitute it into this equation to find the corresponding value of $x$. Understanding this relationship is crucial in algebra, as it allows us to solve for unknowns and analyze how variables interact with each other.

Determining the Value of x When b = 3

Now that we have the expression for $x$ in terms of $b$, we can easily find the value of $x$ when $b$ is equal to 3. This involves a simple substitution. We replace $b$ with 3 in the equation:

x=−3bx = -\frac{3}{b}

Substituting $b = 3$, we get:

x=−33x = -\frac{3}{3}

This is a straightforward division. 3 divided by 3 is 1, so we have:

x=−1x = -1

Therefore, when $b = 3$, the value of $x$ is $-1$. This result illustrates the practical application of the algebraic solution we found earlier. By substituting a specific value for $b$, we were able to determine the corresponding value of $x$. This process is fundamental in many areas of mathematics and science, where equations are used to model real-world phenomena. Understanding how to solve for variables and substitute values is a key skill for anyone working with quantitative information.

Importance of Understanding Algebraic Solutions

The ability to solve algebraic equations like the one presented here is a cornerstone of mathematical literacy. It goes beyond simply finding a numerical answer; it's about understanding the relationships between variables and the logical steps required to manipulate equations. The process of solving for $x$ in terms of $b$ allows us to see the general relationship between these two variables, rather than just a single solution. This is particularly useful in scenarios where $b$ might change, and we need to quickly determine the corresponding value of $x$.

Furthermore, the step-by-step approach we used – distributing, isolating terms, and performing inverse operations – is a pattern that applies to a wide range of algebraic problems. By mastering these techniques, you can tackle more complex equations and mathematical challenges with confidence. The specific example of $-2(bx - 5) = 16$ may seem simple, but it encapsulates the core principles of algebraic manipulation. The ability to confidently and accurately solve these types of equations is critical for advanced mathematical studies and practical applications in science, engineering, and economics.

Conclusion

In summary, we have successfully solved the equation $-2(bx - 5) = 16$ to find the value of $x$ in terms of $b$, which is $x = -\frac{3}{b}$. We then calculated the value of $x$ when $b = 3$, finding that $x = -1$. This exercise demonstrates the fundamental principles of algebraic equation solving, including distribution, isolating variables, and using inverse operations. These skills are essential for anyone pursuing further studies in mathematics or related fields. By understanding these concepts, you can approach more complex problems with a solid foundation and a clear methodology.

Mastering algebra requires practice and a thorough understanding of the underlying principles. By working through examples like this, you can build your confidence and develop the problem-solving skills necessary to succeed in mathematics and beyond. Remember to always double-check your work and ensure that your solutions make sense in the context of the problem. Algebra is not just a set of rules; it's a powerful tool for understanding and modeling the world around us.