Solving The Algebraic Equation (15x+65)(1.2-0.3x)=0 A Comprehensive Guide

Hey guys! Today, we're diving into solving an algebraic equation that might look a bit intimidating at first glance: (15x+65)(1.2-0.3x)=0. But don't worry, we'll break it down step-by-step so it becomes super clear and easy to understand. Algebra might seem like a puzzle sometimes, but with the right approach, you'll be solving equations like a pro in no time. So, let's get started and unravel this equation together!

Understanding the Zero Product Property

Before we jump into the nitty-gritty, let's talk about a fundamental concept in algebra called the Zero Product Property. This property is the key to unlocking solutions for many equations, especially those that are factored, like the one we have here. Simply put, the Zero Product Property states that if the product of two or more factors is zero, then at least one of those factors must be zero. This might sound a bit technical, but it’s actually quite intuitive. Think about it: the only way to get zero when multiplying numbers is if one of the numbers you're multiplying is zero.

So, how does this apply to our equation (15x+65)(1.2-0.3x)=0? Well, we have two factors here: (15x+65) and (1.2-0.3x). According to the Zero Product Property, for this entire expression to equal zero, either (15x+65) must be zero, or (1.2-0.3x) must be zero, or both. This gives us a clear path forward: we can set each factor equal to zero and solve for x individually. This is a common and very effective technique in algebra, and mastering it will help you tackle a wide range of problems. Now that we understand the underlying principle, let's get into the actual steps of solving our equation. Remember, algebra is all about breaking down complex problems into simpler, manageable parts. And with each step we take, we'll get closer to finding our solution. So, keep this principle in mind as we move forward, and you’ll find the process much smoother and more enjoyable!

Step-by-Step Solution

Okay, let's get into the heart of the problem and solve the equation (15x + 65)(1.2 - 0.3x) = 0 step-by-step. Remember that Zero Product Property we talked about? This is where it comes into play. We need to consider each factor separately and set them equal to zero.

1. Setting up the Equations

First, we'll take the first factor, (15x + 65), and set it equal to zero. This gives us our first equation: 15x + 65 = 0. Next, we'll do the same with the second factor, (1.2 - 0.3x), which gives us the second equation: 1.2 - 0.3x = 0. Now we have two separate, simpler equations to solve. This is a classic technique in algebra: breaking down a complex problem into smaller, more manageable parts. By tackling each factor individually, we can systematically find the values of x that make the original equation true. It's like having a big puzzle and sorting the pieces into smaller groups before putting it all together. So, remember this approach as we move forward – it’s a powerful tool for solving equations.

2. Solving the First Equation: 15x + 65 = 0

Now, let's focus on the first equation: 15x + 65 = 0. Our goal here is to isolate x on one side of the equation. To do this, we'll start by subtracting 65 from both sides. This maintains the balance of the equation and gets us closer to our goal. Subtracting 65 from both sides gives us 15x = -65. Next, to completely isolate x, we need to get rid of the 15 that's multiplying it. We do this by dividing both sides of the equation by 15. Dividing both sides by 15, we get x = -65 / 15. Now, let's simplify this fraction. Both 65 and 15 are divisible by 5, so we can reduce the fraction to its simplest form. Dividing both the numerator and the denominator by 5, we get x = -13 / 3. So, one of the solutions to our original equation is x = -13 / 3. See how we systematically worked through the equation, step-by-step, to isolate x? This is the essence of solving algebraic equations. Now, let's move on to the second equation and see what other solution we can find.

3. Solving the Second Equation: 1.2 - 0.3x = 0

Alright, let's tackle the second equation: 1.2 - 0.3x = 0. Just like before, our aim is to isolate x. This time, let's start by subtracting 1.2 from both sides of the equation. This will help us get the term with x by itself. Subtracting 1.2 from both sides gives us -0.3x = -1.2. Now, we need to get rid of the -0.3 that's multiplying x. To do this, we'll divide both sides of the equation by -0.3. Remember, dividing by a negative number will change the sign. Dividing both sides by -0.3, we get x = -1.2 / -0.3. Now, let's simplify this. A negative number divided by a negative number is positive, so we have x = 1.2 / 0.3. To make this easier to calculate, we can multiply both the numerator and the denominator by 10 to get rid of the decimals. This gives us x = 12 / 3. Finally, we can simplify this fraction by dividing 12 by 3, which gives us x = 4. So, the second solution to our original equation is x = 4. We've now found both possible values of x that make the equation true. Isn't it satisfying to see how each step brings us closer to the solution? Now, let’s summarize our findings.

Solutions to the Equation

Okay, we've worked through the steps and found the solutions to the equation (15x + 65)(1.2 - 0.3x) = 0. Let's recap what we discovered. We found two possible values for x that make the equation true. The first solution we found by setting the factor (15x + 65) equal to zero was x = -13/3. This is a negative fraction, which is perfectly acceptable as a solution. The second solution we found by setting the factor (1.2 - 0.3x) equal to zero was x = 4. This is a positive whole number. So, to summarize, the solutions to the equation are x = -13/3 and x = 4. These are the only two values of x that will make the equation equal to zero. It’s like we’ve found the secret keys that unlock the mystery of this equation! Now, to be absolutely sure, it's always a good idea to check our solutions. Let's see how we can do that.

Checking the Solutions

Alright, we've found our solutions, but it's always a smart move to double-check our work. This is where we verify if our solutions, x = -13/3 and x = 4, actually make the original equation (15x + 65)(1.2 - 0.3x) = 0 true. There’s nothing quite like the peace of mind you get from knowing you’ve got the right answer! To check our solutions, we'll substitute each value of x back into the original equation and see if the equation holds true (i.e., if both sides are equal). This is a straightforward process, and it’s a great way to ensure accuracy in algebra. It's like being a detective and confirming your case with solid evidence. So, let's put on our detective hats and get started!

1. Checking x = -13/3

Let's start by plugging x = -13/3 into our equation (15x + 65)(1.2 - 0.3x) = 0. We'll replace every instance of x with -13/3. This gives us (15(-13/3) + 65)(1.2 - 0.3(-13/3)). Now, let's simplify this expression step by step. First, let’s simplify the term 15(-13/3). We can think of 15 as 15/1, so we have (15/1)*(-13/3). Multiplying the numerators and the denominators, we get -195/3. This simplifies to -65. So, the first factor becomes (-65 + 65), which is 0. Now, let's look at the second factor: (1.2 - 0.3(-13/3)). First, we multiply 0.3(-13/3)*. This is equal to -3.9/3, which simplifies to -1.3. So, the second factor becomes (1.2 - (-1.3)), which is 1.2 + 1.3, and that equals 2.5. Now, we have (0)(2.5). Since anything multiplied by zero is zero, the entire expression equals 0. This confirms that x = -13/3 is indeed a solution to the equation. We’ve successfully checked one solution, and it holds true! Now, let's move on to the second solution and see if it also checks out.

2. Checking x = 4

Now, let's check our second solution, x = 4. We'll substitute x = 4 into the original equation (15x + 65)(1.2 - 0.3x) = 0. This gives us (15(4) + 65)(1.2 - 0.3(4)). Let's simplify this step by step. First, we simplify 15(4), which equals 60. So, the first factor becomes (60 + 65), which is 125. Next, let's look at the second factor: (1.2 - 0.3(4)). First, we multiply 0.3(4)*, which equals 1.2. So, the second factor becomes (1.2 - 1.2), which is 0. Now, we have (125)(0). Just like before, anything multiplied by zero is zero, so the entire expression equals 0. This confirms that x = 4 is also a solution to our equation. We've successfully checked both solutions, and they both hold true! This gives us great confidence in our answers.

Conclusion

Awesome! We've successfully solved the algebraic equation (15x + 65)(1.2 - 0.3x) = 0. We found two solutions: x = -13/3 and x = 4. We also took the extra step to check our solutions, ensuring they are correct. Remember, the key to solving equations like this is to understand the Zero Product Property and to break the problem down into smaller, manageable steps. By setting each factor equal to zero and solving for x, we systematically found the values that make the equation true. Algebra can seem challenging at times, but with practice and a clear understanding of the principles, you can tackle even the most complex equations. Keep practicing, and you'll become an algebra whiz in no time! And remember, if you ever get stuck, don't hesitate to review the steps and techniques we've discussed today. You've got this!