Midpoint On A Line: Finding The Equation M Lies On
Let's dive into a fun math problem involving points, lines, and midpoints! This is a classic coordinate geometry question, and we're going to break it down step by step so you can understand exactly how to solve it. We'll explore the concepts of coordinate geometry, equations of lines, and the midpoint formula. So, grab your thinking caps, and let's get started!
Understanding the Problem
The problem states that we have a point P on the graph of the line y = 5x + 3. We also have another point Q with coordinates (3, -2). The point M is the midpoint of the line segment connecting P and Q. Our mission, should we choose to accept it, is to figure out which line M must lie on. This involves understanding how the coordinates of a midpoint relate to the coordinates of the endpoints and how that relationship translates into an equation of a line.
When tackling coordinate geometry problems like this, it's helpful to visualize what's going on. Imagine a line on a graph (y = 5x + 3), and a point Q sitting somewhere off that line. Now, picture a point P moving along the line y = 5x + 3. As P moves, the midpoint M between P and Q will also move, tracing out its own path. Our goal is to find the equation that describes this path of M.
The key concepts we'll use here are the equation of a line (y = mx + c), which describes the relationship between x and y coordinates for points on the line, and the midpoint formula, which helps us find the coordinates of the midpoint given the coordinates of the endpoints. We'll also use a little bit of algebraic manipulation to get our final answer. The beauty of coordinate geometry lies in its ability to translate geometric relationships into algebraic equations, allowing us to solve problems using the power of algebra.
Finding the Midpoint
To solve this, let's first represent the coordinates of point P in terms of a variable. Since P lies on the line y = 5x + 3, we can say the coordinates of P are (x, 5x + 3). This is a crucial step because it allows us to express the coordinates of P using a single variable, x, making the subsequent calculations much simpler. Remember, any point on the line y = 5x + 3 will have its y-coordinate equal to 5 times its x-coordinate plus 3. So, by representing the x-coordinate of P as x, we can automatically express its y-coordinate as 5x + 3.
Now, let's use the midpoint formula to find the coordinates of M, the midpoint of PQ. The midpoint formula states that the coordinates of the midpoint between two points (x₁, y₁) and (x₂, y₂) are (( x₁ + x₂) / 2, (y₁ + y₂) / 2). This formula is derived from the simple idea of averaging the x-coordinates and the y-coordinates of the two points. It's a fundamental tool in coordinate geometry and is used extensively in various problems.
Applying the midpoint formula to points P (x, 5x + 3) and Q (3, -2), we get the coordinates of M as: M = ((x + 3) / 2, (5x + 3 + (-2)) / 2) which simplifies to M = ((x + 3) / 2, (5x + 1) / 2). So, we've now expressed the coordinates of M in terms of x. The x-coordinate of M is (x + 3) / 2, and the y-coordinate of M is (5x + 1) / 2. This is a significant step forward because it connects the coordinates of M back to the variable x, which is related to the position of point P on the line y = 5x + 3.
Determining the Line M Lies On
Let's denote the coordinates of M as (xM, yM). From our previous calculation, we know that xM = (x + 3) / 2 and yM = (5x + 1) / 2. Our goal now is to eliminate the variable x and find a direct relationship between xM and yM. This relationship will give us the equation of the line that M lies on.
First, let's solve the equation xM = (x + 3) / 2 for x. Multiplying both sides by 2, we get 2xM = x + 3. Subtracting 3 from both sides gives us x = 2xM - 3. This is a crucial step because we've now expressed x in terms of xM. We can substitute this expression for x into the equation for yM to eliminate x altogether.
Now, substitute x = 2xM - 3 into the equation yM = (5x + 1) / 2. This gives us yM = (5(2xM - 3) + 1) / 2. Expanding the expression inside the parentheses, we get yM = (10xM - 15 + 1) / 2. Simplifying further, we have yM = (10xM - 14) / 2. Finally, dividing both terms in the numerator by 2, we arrive at yM = 5xM - 7. And there you have it – the equation of the line that M lies on!
The Answer
The equation yM = 5xM - 7 represents the line on which the midpoint M must lie. Looking back at the options provided, we see that this corresponds to option (2). So, the answer is y = 5x - 7. This equation tells us that the y-coordinate of M is always 5 times its x-coordinate minus 7, regardless of the position of point P on the line y = 5x + 3. This elegant relationship is a direct consequence of the midpoint formula and the equation of the line on which P lies.
Key Takeaways
This problem beautifully illustrates the power of coordinate geometry in solving geometric problems using algebraic techniques. The key takeaways from this problem are:
- Representing points on a line: If a point lies on a line, you can express its coordinates in terms of a single variable using the equation of the line.
- The midpoint formula: This formula is essential for finding the midpoint of a line segment given the coordinates of its endpoints.
- Eliminating variables: When you have two equations with three variables, you can often eliminate one variable to find a direct relationship between the other two.
- Connecting geometry and algebra: Coordinate geometry allows us to translate geometric concepts into algebraic equations, which can then be solved using algebraic methods.
By understanding these concepts and techniques, you'll be well-equipped to tackle a wide range of coordinate geometry problems. Remember, practice makes perfect, so keep solving problems and you'll become a coordinate geometry master in no time!
In conclusion, this problem demonstrates how a combination of geometric understanding and algebraic manipulation can lead to a solution. By carefully applying the midpoint formula and eliminating variables, we were able to find the equation of the line on which the midpoint M lies. So next time you encounter a similar problem, remember these steps, and you'll be well on your way to solving it!
