Solving Equations A Step-by-Step Guide

Hey everyone! Today, we're going to dive deep into solving a fascinating equation. So, buckle up and let's get started!

Understanding the Equation

Before we jump into the solution, let's first understand the equation we're dealing with. The equation is:

$\frac{5.5 x}{x^2}=\frac{35+7 x}{6 x+30}$

This equation looks a bit complex, right? But don't worry, we'll break it down step by step. Our goal is to find the value(s) of 'x' that make this equation true. To do that, we'll use some fundamental algebraic principles and techniques.

When it comes to equations like this, where we have fractions, it's crucial to be aware of potential pitfalls. One of the first things we should consider is the denominator. We need to make sure that the denominators are not equal to zero, as division by zero is undefined. So, in this equation, we have two denominators: x^2 and 6x + 30. We need to find the values of 'x' that make these denominators zero and exclude them from our possible solutions.

Let's start with x^2. It's clear that x^2 equals zero when x = 0. So, x = 0 is one value we need to exclude. Now, let's consider the second denominator, 6x + 30. To find the value of 'x' that makes this zero, we can set up the equation 6x + 30 = 0. Solving for 'x', we get:

6x = -30

x = -5

So, x = -5 is another value we need to exclude. These excluded values are called restrictions or undefined points because they would make the original equation undefined. Identifying these restrictions is an essential first step because it helps us avoid incorrect solutions later on. By excluding x = 0 and x = -5, we ensure that our algebraic manipulations are valid, and our final solutions will be correct. With these restrictions in mind, we can proceed with simplifying and solving the equation, knowing that our final answer must not include these values.

Simplifying the Equation

Now that we understand the equation and the restrictions on 'x', let's simplify it. Simplifying an equation makes it easier to solve. One of the most effective ways to simplify this equation is to eliminate the fractions. To do this, we can multiply both sides of the equation by the least common denominator (LCD) of the fractions. The denominators in our equation are x^2 and 6x + 30. To find the LCD, we first need to factor the denominators completely.

The first denominator, x^2, is already in its simplest form. The second denominator, 6x + 30, can be factored by taking out the common factor of 6, which gives us 6(x + 5). Now, we can identify the LCD. The LCD must include all unique factors from both denominators. We have x^2 from the first denominator and 6(x + 5) from the second. So, the LCD is 6x^2(x + 5). This is because it includes the factor x raised to the highest power (which is 2), the constant 6, and the factor (x + 5).

Multiplying both sides of the original equation by this LCD will clear the fractions. This gives us:

6x^2(x + 5) * [5.5x / x^2] = 6x^2(x + 5) * [(35 + 7x) / (6x + 30)]

On the left side, the x^2 term in the denominator cancels out with the x^2 in the LCD, leaving us with:

6(x + 5) * 5.5x

On the right side, the 6(x + 5) term in the denominator cancels out with the 6(x + 5) in the LCD, leaving us with:

x^2 * (35 + 7x)

So, our equation now looks like this:

6(x + 5) * 5.5x = x^2 * (35 + 7x)

This simplified equation is much easier to work with. The next step is to expand both sides and further simplify to bring all terms to one side, setting the equation to zero. This will allow us to solve for 'x' using techniques like factoring or the quadratic formula.

Solving for x

With our simplified equation in hand, the next step is to solve for 'x'. Remember, our simplified equation looks like this:

6(x + 5) * 5.5x = x^2 * (35 + 7x)

To solve this, we'll first expand both sides of the equation. Let's start with the left side. We have:

6(x + 5) * 5.5x = 6 * 5.5x * (x + 5) = 33x(x + 5)

Now, distribute the 33x across the (x + 5):

33x(x + 5) = 33x^2 + 165x

So, the left side of the equation simplifies to 33x^2 + 165x. Now, let's move to the right side of the equation, which is:

x^2 * (35 + 7x)

Distribute the x^2 across the (35 + 7x):

x^2 * (35 + 7x) = 35x^2 + 7x^3

Now, our equation looks like this:

33x^2 + 165x = 35x^2 + 7x^3

To solve for 'x', we need to set the equation to zero. Let's move all terms to one side. We'll subtract 33x^2 and 165x from both sides:

0 = 35x^2 + 7x^3 - 33x^2 - 165x

Combine like terms:

0 = 7x^3 + 2x^2 - 165x

Now we have a cubic equation. To solve this, we can try factoring. First, notice that 'x' is a common factor in all terms. So, we can factor out an 'x':

0 = x(7x^2 + 2x - 165)

This gives us one solution immediately: x = 0. However, remember from our earlier discussion that x = 0 is a restriction, as it makes the original equation undefined. So, we discard this solution.

Now, we need to solve the quadratic equation inside the parentheses:

7x^2 + 2x - 165 = 0

This quadratic equation doesn't seem easily factorable, so we can use the quadratic formula to find the solutions for 'x'. The quadratic formula is:

x = [-b ± √(b^2 - 4ac)] / (2a)

In our equation, a = 7, b = 2, and c = -165. Plug these values into the quadratic formula:

x = [-2 ± √(2^2 - 4 * 7 * -165)] / (2 * 7)

x = [-2 ± √(4 + 4620)] / 14

x = [-2 ± √4624] / 14

x = [-2 ± 68] / 14

This gives us two possible solutions:

x = (-2 + 68) / 14 = 66 / 14 = 33 / 7

x = (-2 - 68) / 14 = -70 / 14 = -5

However, we also found earlier that x = -5 is a restriction. So, we discard this solution as well. This leaves us with only one valid solution: x = 33 / 7. Therefore, by simplifying the equation, factoring, applying the quadratic formula, and considering the restrictions on 'x', we've successfully found the solution to the equation.

Verifying the Solution

We've arrived at a solution, but it's always a good practice to verify our answer. Verifying the solution means plugging it back into the original equation to ensure it holds true. This step is crucial because it helps us catch any potential errors we might have made during the solving process. Our original equation is:

$\frac{5.5 x}{x^2}=\frac{35+7 x}{6 x+30}$

And our solution is x = 33 / 7. To verify, we'll substitute x = 33 / 7 into both sides of the equation and see if they are equal.

Let's start with the left side:

$\frac{5. 5 x}{x^2} = \frac{5.5 * (33 / 7)}{(33 / 7)^2}$

Simplify the numerator:

5.5 * (33 / 7) = (11 / 2) * (33 / 7) = 363 / 14

Simplify the denominator:

(33 / 7)^2 = 1089 / 49

Now, divide the numerator by the denominator:

(363 / 14) / (1089 / 49) = (363 / 14) * (49 / 1089)

To simplify this fraction, we can cancel out common factors. Notice that 363 and 1089 have a common factor of 363, and 14 and 49 have a common factor of 7. So, we can simplify:

(363 / 14) * (49 / 1089) = (1 / 2) * (7 / 3) = 7 / 6

So, the left side of the equation equals 7 / 6 when x = 33 / 7.

Now, let's evaluate the right side of the equation:

$\frac{35+7 x}{6 x+30} = \frac{35 + 7 * (33 / 7)}{6 * (33 / 7) + 30}$

Simplify the numerator:

35 + 7 * (33 / 7) = 35 + 33 = 68

Simplify the denominator:

6 * (33 / 7) + 30 = 198 / 7 + 30

To add these terms, we need a common denominator, which is 7. So, we rewrite 30 as 210 / 7:

198 / 7 + 210 / 7 = 408 / 7

Now, divide the numerator by the denominator:

68 / (408 / 7) = 68 * (7 / 408)

Again, we can simplify this fraction by canceling out common factors. Notice that 68 and 408 have a common factor of 68. So, we can simplify:

68 * (7 / 408) = 1 * (7 / 6) = 7 / 6

So, the right side of the equation also equals 7 / 6 when x = 33 / 7.

Since both sides of the equation are equal when x = 33 / 7, our solution is verified. This step confirms that our algebraic manipulations were correct, and we have found the correct value for 'x' that satisfies the original equation. Through this process of verification, we gain confidence in our solution and ensure the accuracy of our work. Great job, guys! You've successfully verified the solution to the equation.

Conclusion

Solving equations can be challenging, but by breaking them down step by step, we can find the solutions. Remember to always check for restrictions and verify your solutions! So, to wrap things up, let's recap the key steps we took to solve this equation.

First, we started by understanding the equation and identifying any restrictions on 'x'. We recognized that the denominators could not be zero, which led us to exclude x = 0 and x = -5 as possible solutions. Identifying these restrictions was a crucial first step because it helped us avoid incorrect solutions later on. Next, we simplified the equation by multiplying both sides by the least common denominator (LCD). This cleared the fractions and made the equation easier to work with. The simplified equation allowed us to expand the terms and rearrange them into a standard form.

After simplifying, we solved for 'x' by factoring and applying the quadratic formula. Factoring out a common factor of 'x' allowed us to identify one potential solution, but we remembered our earlier restriction and discarded it. The remaining quadratic equation required the use of the quadratic formula, which gave us two possible solutions. However, one of these solutions was also a restriction, so we discarded it as well. This left us with only one valid solution. Finally, we verified our solution by plugging it back into the original equation. This step confirmed that our algebraic manipulations were correct and that our solution satisfied the equation. Verification is an essential part of the problem-solving process because it ensures the accuracy of our work.

Through this process, we not only found the solution to the equation but also reinforced our understanding of algebraic principles and techniques. Solving equations is a fundamental skill in mathematics, and mastering it opens the door to more advanced topics and applications. So, keep practicing, guys, and you'll become even more proficient at solving equations. I hope this explanation has been helpful in clarifying the steps involved in solving equations. Happy problem-solving!