Como Encontrar O Ponto De Interseção Entre F(x) = 5x - 3 E A(x) = -1/4x + 3

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Hey guys! Ever wondered how to find where two lines meet? That's what we're diving into today! We're going to figure out the intersection point of two functions, f(x) = 5x - 3 and a(x) = -1/4x + 3. It might sound intimidating, but trust me, it's like solving a puzzle, and we'll break it down step-by-step. So, buckle up, grab your thinking caps, and let's get started!

Understanding the Concept of Intersection Points

Before we jump into the math, let's make sure we're all on the same page. The intersection point is simply the point where two or more lines or curves cross each other on a graph. At this point, the functions share the same x and y values. Think of it like a meeting place for the two functions. To find this meeting place, we need to solve the system of equations formed by the functions. This means finding the values of x and y that satisfy both equations simultaneously. In simpler terms, we're looking for the x value that, when plugged into both equations, gives us the same y value. This shared x and y value is the coordinate of the intersection point. Finding the intersection point is a fundamental concept in mathematics with applications ranging from basic algebra to advanced calculus and beyond. In real-world scenarios, intersection points can represent break-even points in business, equilibrium points in economics, or collision points in physics. Understanding how to calculate these points is therefore essential for problem-solving across various disciplines. For example, businesses use this concept to determine the sales volume required to cover costs, while economists use it to analyze market supply and demand dynamics. In physics, understanding intersections is crucial for predicting the trajectories of moving objects and determining potential collisions. This concept is also vital in computer graphics, where finding the intersection points of lines and curves is necessary for rendering images and creating realistic simulations. Whether you are working with linear equations, quadratic equations, or more complex functions, the basic principle remains the same: set the equations equal to each other and solve for the variables. This process reveals the points where the functions have the same output, providing valuable insights into their relationship and behavior. This method allows us to compare different models and make informed decisions based on the points where they align or diverge. Mastering the techniques for finding intersection points opens up a wide array of analytical tools and practical applications, making it a cornerstone skill in mathematics and its related fields.

Setting Up the System of Equations

Okay, now that we've got the concept down, let's get our hands dirty with the math! Our first step is to set up the system of equations. We have two functions:

  • f(x) = 5x - 3
  • a(x) = -1/4x + 3

To find the intersection point, we need to find the x value where f(x) and a(x) are equal. So, we'll set them equal to each other: 5x - 3 = -1/4x + 3. This equation represents the condition where both functions have the same y value for a given x value, which is exactly what we're looking for in an intersection point. By equating the two functions, we transform the problem of finding the intersection point into a problem of solving a single equation. This approach is a common and powerful technique in mathematics, allowing us to use algebraic methods to find geometric solutions. The next step involves manipulating this equation to isolate the variable x, which will give us the x-coordinate of the intersection. Understanding this setup is crucial because it forms the foundation for the rest of the solution process. Once we have the x-coordinate, we can substitute it back into either of the original equations to find the corresponding y-coordinate. This process highlights the interconnectedness of algebraic and graphical representations in mathematics. By solving the equation 5x - 3 = -1/4x + 3, we are essentially finding the x-coordinate of the point where the graphs of the two functions intersect. This algebraic solution provides a precise way to determine the location of this point, which can be visualized graphically as the meeting point of the two lines. Setting up the equation correctly is essential for arriving at the correct solution. A mistake in this initial step can lead to incorrect results, underscoring the importance of careful attention to detail in mathematical problem-solving. Once the equation is set up, the next challenge is to solve it efficiently and accurately, using algebraic techniques such as combining like terms, isolating variables, and simplifying expressions. This step-by-step approach ensures a systematic and reliable method for finding the intersection points of functions.

Solving the Equation

Alright, we've set up the equation: 5x - 3 = -1/4x + 3. Now comes the fun part – solving for x! Our goal is to get x all by itself on one side of the equation. First, let's get rid of that pesky fraction. We can do this by multiplying both sides of the equation by 4. This gives us:

  • 4(5x - 3) = 4(-1/4x + 3)
  • 20x - 12 = -x + 12

Now, let's get all the x terms on one side and the constants on the other. We can add x to both sides:

  • 20x + x - 12 = -x + x + 12
  • 21x - 12 = 12

Next, we'll add 12 to both sides:

  • 21x - 12 + 12 = 12 + 12
  • 21x = 24

Finally, to isolate x, we'll divide both sides by 21:

  • 21x / 21 = 24 / 21
  • x = 24/21

We can simplify this fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 3:

  • x = (24 / 3) / (21 / 3)
  • x = 8/7

So, we've found our x value! It's 8/7. Solving the equation involves a series of algebraic manipulations, each designed to isolate the variable x. These manipulations include distributing multiplication, combining like terms, and using inverse operations to move terms from one side of the equation to the other. The key to successful equation solving is to perform the same operation on both sides, ensuring that the equality is maintained throughout the process. Each step builds upon the previous one, gradually bringing us closer to the solution. The use of parentheses in the initial distribution step is crucial for correctly multiplying the constant by each term inside the parentheses. Mistakes in distribution can lead to incorrect solutions, emphasizing the importance of careful attention to detail. Combining like terms simplifies the equation, making it easier to work with. This step involves adding or subtracting terms that have the same variable and exponent. Inverse operations, such as adding or subtracting the same number from both sides, help to isolate the variable. These operations essentially