Find The Intersection Of Two Lines: Step-by-Step Solution
Introduction
Hey guys! Today, we're diving into a classic math problem: finding the intersection point of two lines. This is a fundamental concept in algebra and geometry, and it pops up in all sorts of real-world applications, from figuring out where two roads cross to optimizing business decisions. We'll tackle this by solving a system of equations. So, buckle up and let's get started!
We will be addressing the intersection of lines, which is a fascinating topic that brings together concepts from both algebra and geometry. When we talk about the intersection point, we're essentially looking for the single point on a graph where two or more lines meet or cross each other. This point is crucial because it represents the unique solution that satisfies the equations of all the lines involved. In simpler terms, it's the one place where the x and y values work for both equations simultaneously. Understanding how to find this point is not only a key skill in mathematics but also has practical applications in various fields, such as engineering, economics, and computer graphics. For instance, engineers might use this concept to determine the point of stability in structural designs, while economists could use it to find the equilibrium point in supply and demand curves. In the realm of computer graphics, understanding intersections is fundamental for rendering 3D scenes and handling collision detection in games. So, as we delve into this topic, remember that we're not just solving equations; we're unlocking a powerful tool that can be applied in many different contexts. The journey of finding the intersection point involves understanding what it means for lines to intersect and then employing algebraic techniques to pinpoint that exact location. We'll be focusing on the system of equations approach, which is one of the most straightforward and universally applicable methods. This involves setting up the equations of the lines and then solving them simultaneously. This may sound a bit technical now, but don't worry; we'll break it down into easy-to-follow steps and explain each part in detail. By the end of this article, you'll be able to confidently tackle these problems and understand the underlying principles at play.
Problem Statement
The problem we're tackling today is: What is the intersection point of the lines represented by the equations y = 3x - 2 and y = 2 - x? To solve this, we need to find the values of x and y that make both equations true at the same time. Let's dive into the solution!
To properly set the stage for solving this problem, it's important to thoroughly understand what the question is asking. When we are presented with two equations, y = 3x - 2 and y = 2 - x, we are essentially given a pair of lines in a two-dimensional coordinate system. Each of these equations represents a straight line when plotted on a graph. The equation y = 3x - 2 is in slope-intercept form, where 3 is the slope and -2 is the y-intercept. This means that the line rises 3 units for every 1 unit it moves to the right, and it crosses the y-axis at the point (0, -2). The second equation, y = 2 - x, can be rewritten as y = -x + 2, which also puts it in slope-intercept form. Here, the slope is -1, indicating that the line falls 1 unit for every 1 unit it moves to the right, and the y-intercept is 2, meaning it crosses the y-axis at the point (0, 2). The intersection point we're trying to find is the unique location where these two lines cross each other on the graph. At this point, the x and y coordinates will satisfy both equations simultaneously. This is a critical concept because it means that if we can find a pair of values (x, y) that work for both equations, we've successfully identified the intersection. The task now is to employ algebraic methods to discover these values. The beauty of this problem lies in its visual interpretability. If we were to draw these lines on a graph, the intersection point would be where they visually cross. However, solving the system of equations algebraically allows us to find this point precisely without relying on a visual representation, which might not always be accurate. Moreover, this algebraic approach is scalable and can be used for more complex systems of equations where graphical methods might not be practical. So, we're not just solving for the sake of solving; we're honing a powerful mathematical technique that has broader applicability. The problem can be approached through several methods, but we'll be focusing on the substitution method in this explanation, as it's a straightforward way to handle systems of equations where one variable is already isolated in both equations. Before we jump into the solution, take a moment to visualize what's happening. We have two lines, each with its own slope and y-intercept, and we're on a quest to find the one point where they meet. This visual context can often make the algebraic steps more intuitive and easier to remember.
Solving the System of Equations
Okay, let's get our hands dirty and solve this thing! We have two equations:
- y = 3x - 2
- y = 2 - x
Since both equations are already solved for y, we can use the substitution method. This means we can set the right-hand sides of the equations equal to each other:
3x - 2 = 2 - x
Now, let's solve for x. Add x to both sides:
4x - 2 = 2
Next, add 2 to both sides:
4x = 4
Finally, divide both sides by 4:
x = 1
Awesome! We've found the x-coordinate of the intersection point. Now, to find the y-coordinate, we can plug this value of x into either of our original equations. Let's use the second equation, y = 2 - x:
y = 2 - 1 y = 1
So, the y-coordinate is also 1. This means the intersection point is (1, 1).
Let's break down the solution process step by step to ensure we understand each move and why it's made. The core of solving a system of equations is to find the values of the variables that satisfy all equations in the system simultaneously. In our case, we have two equations and two variables (x and y), which is a common scenario in algebra. The method of substitution is particularly effective when one or more equations are already solved for one variable in terms of the other, which is precisely what we have here. Both equations are explicitly solved for y, making substitution a natural choice. The initial move in this method is to equate the expressions that y is equal to. This is based on the transitive property of equality: if a = b and b = c, then a = c. In our case, since y is equal to both 3x - 2 and 2 - x, we can say that 3x - 2 is equal to 2 - x. This gives us a new equation with only one variable, x, which we can then solve. The process of solving for x involves a series of algebraic manipulations aimed at isolating x on one side of the equation. We first add x to both sides of the equation to eliminate the -x term on the right side. This is a crucial step because it consolidates the x terms on one side, making the equation simpler to solve. Next, we add 2 to both sides to isolate the term with x. This is another application of the addition property of equality, which states that adding the same number to both sides of an equation preserves the equality. Finally, we divide both sides by 4 to solve for x. This is the division property of equality, and it's the final step in isolating x. We find that x = 1, which is the x-coordinate of our intersection point. Now that we have the value of x, we need to find the corresponding value of y. This is where the substitution part of the method comes full circle. We can substitute the value of x into either of the original equations to solve for y. The choice of which equation to use is arbitrary, as both will yield the same result if the x value is correct. However, it's often strategic to choose the simpler equation to minimize the chances of computational errors. In our case, y = 2 - x appears slightly simpler, so we substitute x = 1 into it. This gives us y = 2 - 1, which simplifies to y = 1. Therefore, the y-coordinate of the intersection point is also 1. Combining the x and y values, we find that the intersection point is (1, 1). This is the solution to our system of equations and the point where the two lines intersect on the graph. To be absolutely sure of our answer, it's always a good practice to check it by substituting both x and y values back into both original equations. If both equations hold true, we can confidently say that we've found the correct intersection point.
Verification
To make sure we've nailed it, let's verify our solution. We'll plug the point (1, 1) back into our original equations:
- y = 3x - 2 --> 1 = 3(1) - 2 --> 1 = 1 (Yep!)
- y = 2 - x --> 1 = 2 - 1 --> 1 = 1 (Double yep!)
Both equations hold true, so (1, 1) is indeed the intersection point.
Verifying our solution is a crucial step in any mathematical problem, especially when dealing with systems of equations. It's our way of ensuring that the values we've calculated for x and y not only solve the individual equations but also work together to satisfy the entire system. This process involves taking the x and y values we found, which in our case is the point (1, 1), and substituting them back into the original equations. If the equations hold true after the substitution, it confirms that our solution is correct. Let's walk through this process in detail. Our first equation is y = 3x - 2. We'll substitute x = 1 and y = 1 into this equation. This gives us 1 = 3(1) - 2. Simplifying the right-hand side, we get 1 = 3 - 2, which further simplifies to 1 = 1. This is a true statement, which means the point (1, 1) satisfies the first equation. Next, we'll move on to the second equation, which is y = 2 - x. Again, we substitute x = 1 and y = 1 into this equation. This results in 1 = 2 - 1. Simplifying the right-hand side, we get 1 = 1, which is also a true statement. This confirms that the point (1, 1) satisfies the second equation as well. Since the point (1, 1) satisfies both equations, we can confidently conclude that it is the correct solution to the system of equations. This verification step is not just a formality; it's a robust way to catch any errors that might have occurred during the solving process. For instance, if we had made a mistake in our algebraic manipulations and arrived at an incorrect solution, the verification step would likely reveal this discrepancy. The substituted values would not make the equations true, signaling that we need to revisit our steps. Moreover, the verification process reinforces the concept of what it means to solve a system of equations. It highlights that the solution must work for all equations simultaneously, not just one or the other. This understanding is crucial for tackling more complex problems in the future. In some cases, you might encounter systems of equations that have no solution or infinitely many solutions. The verification step can help you identify such scenarios as well. If, after substituting the values, you arrive at a contradiction (e.g., 0 = 1), it indicates that the system has no solution. If the equations are essentially the same, you might find that any point on the line satisfies both equations, indicating infinitely many solutions. So, always remember to verify your solutions. It's a simple yet powerful tool that ensures accuracy and deepens your understanding of the underlying mathematical principles.
Conclusion
There you have it, folks! The intersection point of the lines y = 3x - 2 and y = 2 - x is (1, 1). We found this by using the substitution method, which is a super useful technique for solving systems of equations. Hope you had fun solving this with me!
Wrapping up, let's take a moment to reflect on what we've accomplished and the broader implications of finding the intersection point of two lines. Throughout this discussion, we've not only solved a specific problem but also reinforced key concepts in algebra and analytical geometry. We've seen how a system of equations can represent a geometrical situation, specifically the intersection of lines, and how solving this system gives us the precise coordinates where these lines meet. The method we used, substitution, is a powerful algebraic tool that's applicable in a wide range of mathematical contexts. It's not just about plugging numbers into equations; it's about understanding the underlying relationships and manipulating them to find a solution. The process of setting the equations equal to each other, solving for one variable, and then substituting back to find the other variable is a testament to the logical and systematic approach that mathematics encourages. Furthermore, the verification step we took underscores the importance of accuracy and thoroughness in problem-solving. It's a reminder that getting an answer is only half the battle; ensuring that the answer is correct is equally crucial. This habit of verification is invaluable, not just in mathematics but in any field that requires precise and reliable results. But beyond the mechanics of solving this particular problem, there's a larger picture to consider. The concept of intersection extends far beyond lines on a graph. It's a fundamental idea in many areas of science, engineering, economics, and even social sciences. Anytime you're looking for a point where multiple conditions are simultaneously met, you're essentially looking for an intersection. In engineering, it might be the point where stress and strain reach a critical threshold in a structure. In economics, it could be the equilibrium point where supply and demand curves intersect. In computer science, it might be the point where two algorithms have the same output for a given input. Therefore, understanding how to find intersections is a valuable skill that transcends the classroom. It's a skill that empowers you to analyze and solve problems in a variety of real-world contexts. So, as you move forward in your mathematical journey, remember the principles we've discussed today. Remember the importance of understanding the problem, choosing the right method, executing the steps carefully, and always verifying your solution. And remember that mathematics is not just about numbers and equations; it's about developing a way of thinking that can help you navigate the complexities of the world around you.
Options
The correct answer is B) (1, 1).
