Solve For $d$.$\frac{-d}{7d+50} = \frac{1}{d+8}$There May Be 1 Or 2 Solutions.$d = \square$ Or $d = \square$

Introduction

Solving equations involving fractions can be a challenging task, especially when dealing with variables in the denominator. In this article, we will guide you through the process of solving the equation $\frac{-d}{7d+50} = \frac{1}{d+8}$, which may have one or two solutions. We will break down the solution into manageable steps, making it easier to understand and follow along.

Step 1: Cross-Multiply to Eliminate the Fractions

To eliminate the fractions, we will cross-multiply the two sides of the equation. This means multiplying both sides by the denominators of the fractions, which are $(7d+50)$ and $(d+8)$.

$$\frac{-d}{7d+50} = \frac{1}{d+8}$$

Cross-multiplying gives us:

$$-d(d+8) = 1(7d+50)$$

Step 2: Expand and Simplify the Equation

Now, we will expand and simplify the equation by multiplying the terms inside the parentheses.

$$-d(d+8) = 1(7d+50)$$

Expanding the left side gives us:

$$-d^2 - 8d = 7d + 50$$

Step 3: Move All Terms to One Side of the Equation

To solve for $d$, we need to move all the terms to one side of the equation. We will add $d^2 + 8d$ to both sides to get:

$$-d^2 - 8d + 7d + 50 = 0$$

Simplifying the equation gives us:

$$-d^2 - d + 50 = 0$$

Step 4: Multiply Both Sides by -1 to Simplify the Equation

To simplify the equation, we will multiply both sides by -1, which gives us:

$$d^2 + d - 50 = 0$$

Step 5: Factor the Quadratic Equation

Now, we will factor the quadratic equation to find the values of $d$ that satisfy the equation.

$$d^2 + d - 50 = 0$$

Factoring the equation gives us:

$$(d + 10)(d - 5) = 0$$

Step 6: Solve for $d$

To find the values of $d$, we will set each factor equal to zero and solve for $d$.

$$(d + 10) = 0$$

Solving for $d$ gives us:

$$d = -10$$

$$(d - 5) = 0$$

Solving for $d$ gives us:

$$d = 5$$

Conclusion

In this article, we have solved the equation $\frac{-d}{7d+50} = \frac{1}{d+8}$, which may have one or two solutions. We have broken down the solution into manageable steps, making it easier to understand and follow along. The solutions to the equation are $d = -10$ and $d = 5$. We hope this article has provided you with a clear understanding of how to solve equations involving fractions and variables in the denominator.

Final Answer

The final answer is:

$d = \boxed{-10}$ or $d = \boxed{5}$