Solving A*b + 3a + 3b = 14: A Complete Guide

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Hey guys! Ever stumbled upon an equation that looks a bit... intimidating? Well, let's break down one such equation today: a*b + 3a + 3b = 14. It might seem a little tricky at first glance, but trust me, with a few clever steps, we can solve this with ease. This guide will walk you through a step-by-step process to find the possible values for 'a' and 'b'. We'll use a technique called factoring, which is a super handy tool in algebra. So, buckle up, and let's dive in!

Understanding the Equation and Our Goal

First things first, let's understand what we're dealing with. The equation a*b + 3a + 3b = 14 involves two variables, 'a' and 'b', and our goal is to find pairs of numbers (a, b) that satisfy this equation. These numbers could be integers, fractions, or even decimals. The main trick here is to rearrange the equation in a way that we can factor it. Factoring means rewriting the equation as a product of simpler expressions. This will help us isolate 'a' and 'b' and find their possible values. We are looking for numbers that, when plugged into the original equation, make the left side equal to 14. Think of it like a puzzle where we need to find the pieces that fit perfectly. We will manipulate the equation, add and subtract strategically, until we can rewrite it in a more manageable form. Keep in mind that the goal is to simplify and rewrite the equation using a method that will show us the possible values of 'a' and 'b'. We will use factoring by grouping to achieve this. It is a fundamental algebraic technique that allows us to find solutions where the variables interact with each other, and is very useful for this particular problem.

To succeed, we will be using basic algebraic manipulations. First, observe that the equation a*b + 3a + 3b = 14 can be slightly rearranged by grouping terms. The presence of the a*b, 3a, and 3b terms suggests we might be able to factor by grouping. The constant 14 on the right-hand side is a clue. It tells us that after factoring, we'll likely end up with factors that multiply to give us a value that is somehow linked to 14. Keep this in mind. It is also essential to know what kind of numbers we're looking for. In this example, we assume that 'a' and 'b' are real numbers, though the approach works for various other number systems. The strategy will remain the same, although the possible values for 'a' and 'b' might change based on the context.

Step-by-Step Solution: Factoring to the Rescue!

Alright, let's get our hands dirty and solve this thing. The core idea here is to transform the equation into a form that we can factor. We'll use a technique that involves creating a common factor.

  1. Rearrange and Add a Constant: Start with the given equation: a*b + 3a + 3b = 14. Our goal is to factor the left side. Notice that if we could somehow get a term like 9 on the left side, we might be able to factor. To do this, we will add 9 to both sides of the equation. This might seem like a random step, but trust me, there's a method to the madness. Adding 9 to both sides gives us: a*b + 3a + 3b + 9 = 14 + 9, which simplifies to a*b + 3a + 3b + 9 = 23. This step is crucial because it allows us to group terms and ultimately factor the equation.

  2. Factor by Grouping: Now, let's group the terms strategically: a*b + 3a + 3b + 9 = 23. We can rewrite the left side as a*(b + 3) + 3*(b + 3) = 23. Notice that we now have a common factor of (b + 3). So, we can factor out (b + 3) to get (a + 3)*(b + 3) = 23. This is where the magic happens. We have successfully factored the equation!

  3. Identify Factors of the Constant: The equation (a + 3)*(b + 3) = 23 tells us that the product of (a + 3) and (b + 3) equals 23. Since 23 is a prime number, its only factors are 1 and 23 (and -1 and -23). This significantly simplifies our work. We know that (a + 3) and (b + 3) must be factors of 23.

  4. Solve for 'a' and 'b': Now, let's consider the possible factor pairs of 23. We have two main cases:

    • Case 1: (a + 3) = 1 and (b + 3) = 23. Solving for 'a' and 'b', we get a = -2 and b = 20.
    • Case 2: (a + 3) = 23 and (b + 3) = 1. Solving for 'a' and 'b', we get a = 20 and b = -2.
    • Case 3: (a + 3) = -1 and (b + 3) = -23. Solving for 'a' and 'b', we get a = -4 and b = -26.
    • Case 4: (a + 3) = -23 and (b + 3) = -1. Solving for 'a' and 'b', we get a = -26 and b = -4.

  5. Check Your Answers: Always a good idea to plug your solutions back into the original equation a*b + 3a + 3b = 14 to make sure they work. For example, let's check the first solution a = -2 and b = 20: (-2)*20 + 3*(-2) + 3*20 = -40 - 6 + 60 = 14. It checks out! Do the same for the other solutions to confirm they are correct.

Understanding the Solutions and Their Implications

So, we've found a few solutions for the equation a*b + 3a + 3b = 14. These solutions are pairs of numbers (a, b) that satisfy the equation. For this equation, we obtained four sets of solutions: (-2, 20), (20, -2), (-4, -26), and (-26, -4). Each of these represents a valid combination of 'a' and 'b' that, when plugged into the original equation, will make the left side equal to 14. It's worth noting that the order of 'a' and 'b' in a pair matters. The solutions are unique to the equation; they are the only number combinations that will work. Knowing how to verify the results is vital. The more you practice, the more confident you will be in solving similar equations.

These solutions aren't just random numbers; they represent the points where the equation holds true. If you were to graph this equation, these would be specific points on the curve. In this case, the relationship is not a simple linear equation. It's a bit more complex, involving a product of the variables. In real-world applications, equations like this can model various phenomena, from physics to economics. Recognizing patterns and manipulating equations are key skills in mathematics and other scientific fields. It also reinforces the importance of understanding the context of a problem. Are we looking for integer solutions, real number solutions, or something else? This will determine the scope of our search. For example, if we only wanted positive integer solutions, we would only find one solution from all those we have calculated.

Key Takeaways and Further Exploration

Alright, folks, let's wrap things up with some key takeaways. We've successfully solved the equation a*b + 3a + 3b = 14 using the factoring technique. Here's what you should remember:

  • Factoring is Your Friend: Recognizing opportunities to factor equations is a powerful skill in algebra. It simplifies complex expressions into more manageable forms.
  • Adding Constants Can Help: Sometimes, adding a specific constant to both sides of an equation can set you up for factoring. This might not always be immediately obvious, but it's a common trick.
  • Prime Numbers Simplify Things: When you have a product equal to a prime number, your work gets a lot easier because you only need to consider the factors of that prime number.
  • Always Check Your Work: Plugging your solutions back into the original equation is crucial to verify that your answers are correct.

If you enjoyed this, here are a few ideas for further exploration:

  • Practice, practice, practice: Try solving similar equations with different coefficients and constants. The more you practice, the better you'll get.
  • Explore Other Factoring Techniques: There are other factoring techniques, such as difference of squares or factoring quadratics. Learn these, as they will expand your skillset. Find more complex equations to solve, and see if you can apply these methods.
  • Visualize the Equation: If you're into visuals, try graphing the equation. Seeing the solutions on a graph can provide a different perspective and deepen your understanding. You can use online graphing calculators or software like Desmos to easily plot the equation and see the solutions visually.

By following these steps and practicing regularly, you'll become a pro at solving this type of equation. Keep experimenting, and don't be afraid to try new things. Keep practicing, and you'll be solving these equations with ease. You got this! Thanks for joining me today!