Solving Equations: Finding M + N - P

Hey math enthusiasts! Today, we're diving into a cool algebra problem that involves solving an equation and finding the value of an expression. Specifically, we're going to break down the equation 6x - 36 = mx(x - 3) + nx(x + 3) + p(x² - 9) and figure out what m + n - p equals. Sounds interesting, right? Let's get started, guys!

Unpacking the Equation and Our Goal

Alright, let's start by looking closely at our equation. We've got a linear expression on the left side: 6x - 36. On the right side, things are a bit more complicated. We see terms with x, m, n, and p. The goal is to determine the values of m, n, and p that make this equation true for all values of x. Once we know those values, we can easily calculate m + n - p. This type of problem is all about algebraic manipulation and understanding how to deal with different forms of equations. We'll use techniques like expanding, collecting like terms, and comparing coefficients to crack this puzzle. It's like being a detective, except instead of clues, we're working with equations! Get ready to explore the exciting world of algebra.

Now, before we move on, let's make sure we're all on the same page. The equation essentially tells us that the two sides are equal. This must be true no matter what value we plug in for x. Our task is to find the specific values for m, n, and p that ensure this equality holds true. We need to work with the right side of the equation, the one with m, n, and p, and manipulate it so it looks like the left side. Then, by comparing the coefficients of like terms (terms with the same power of x), we can find our unknown values. It's like a matching game! We need to make sure everything on the right side lines up perfectly with everything on the left side.

Step-by-Step Solution: Finding m, n, and p

Okay, guys, let's roll up our sleeves and start solving! The first thing we need to do is expand the right side of the equation. This means we'll multiply out the terms. We'll be using the distributive property, which is like saying we're going to share the love (or the multiplication) across all the terms within the parentheses. The right side of the equation is: mx(x - 3) + nx(x + 3) + p(x² - 9). Let's break it down:

• mx(x - 3) = mx² - 3mx (Multiply mx by both x and -3)
• nx(x + 3) = nx² + 3nx (Multiply nx by both x and 3)
• p(x² - 9) = px² - 9p (Multiply p by both x² and -9)

Now, let's put it all together. The expanded right side becomes: mx² - 3mx + nx² + 3nx + px² - 9p. It looks a bit messy, right? But don't worry, we're going to tidy it up. The next thing we do is collect the like terms. This means we'll group together all the terms that have the same power of x. We have terms with x², terms with x, and constant terms (those without any x). Let's group them:

• x² terms: mx² + nx² + px² = (m + n + p)x²
• x terms: -3mx + 3nx = (-3m + 3n)x
• Constant terms: -9p

So, the simplified right side of the equation is (m + n + p)x² + (-3m + 3n)x - 9p. Now our equation looks like this: 6x - 36 = (m + n + p)x² + (-3m + 3n)x - 9p. See? Much cleaner and easier to work with! Comparing coefficients is the next big step.

Comparing Coefficients: The Key to the Puzzle

Alright, we've got our equation simplified, and now it's time to put on our detective hats and compare the coefficients. Remember, the original equation, 6x - 36 = (m + n + p)x² + (-3m + 3n)x - 9p, must hold true for all values of x. This means that the coefficients of the corresponding terms on both sides of the equation must be equal. Let's break it down by looking at the powers of x:

1. x² terms: On the left side, there is no x² term. This means the coefficient of x² on the right side must be zero. So, we have: m + n + p = 0.
2. x terms: On the left side, the coefficient of x is 6. On the right side, the coefficient of x is (-3m + 3n). Therefore: -3m + 3n = 6.
3. Constant terms: On the left side, the constant term is -36. On the right side, the constant term is -9p. So: -9p = -36.

We now have a system of three equations:

• m + n + p = 0
• -3m + 3n = 6
• -9p = -36

This is great news, guys! We have three equations and three unknowns, which means we can solve for m, n, and p. Let's do it step by step. First, let's solve for p using the third equation: -9p = -36. Dividing both sides by -9, we get p = 4. Now that we know p, we can plug it into the first equation: m + n + 4 = 0, which simplifies to m + n = -4. Next, let's simplify the second equation: -3m + 3n = 6. Dividing both sides by 3, we get -m + n = 2. We now have two equations with two unknowns: m + n = -4 and -m + n = 2. We can solve this system using substitution or elimination. Let's use elimination by adding the two equations together: (m + n) + (-m + n) = -4 + 2. This simplifies to 2n = -2, so n = -1. Finally, we can substitute the value of n back into one of our equations to solve for m. Using m + n = -4, we have m + (-1) = -4, so m = -3. We've got our values, guys! m = -3, n = -1, and p = 4.

Calculating m + n - p: The Final Answer!

We're in the home stretch, guys! We've done all the hard work, and now it's time to find the value of m + n - p. We've already found that:

• m = -3
• n = -1
• p = 4

Now, let's substitute these values into the expression: m + n - p = (-3) + (-1) - 4. Simplifying this gives us: -3 - 1 - 4 = -8. So, the final answer, the value of m + n - p is -8. Awesome! We did it!

Recap and Key Takeaways

Let's take a quick look back at what we've accomplished today. We started with the equation 6x - 36 = mx(x - 3) + nx(x + 3) + p(x² - 9), and our goal was to find the value of m + n - p. We expanded the right side of the equation, collected like terms, and then compared the coefficients of the terms on both sides. This allowed us to create a system of equations, which we solved to find the values of m, n, and p. Finally, we substituted these values into the expression m + n - p and found our answer, which is -8. The key takeaways from this exercise are the importance of algebraic manipulation, understanding how to expand and simplify expressions, the power of comparing coefficients to solve equations, and the ability to work through a system of equations. Practice these skills, guys, and you'll be well on your way to mastering algebra. Keep practicing, and don't be afraid to try new problems! The more you practice, the easier it will become. You've got this!