Line Equation Through Two Points: A Quick Guide
Hey guys! Today, we're diving into a common problem in math: finding the equation of a line when you're given two points. Specifically, we’ll tackle the problem: Given the points A(2, 3) and B(5, 7), what is the general equation of the line that passes through them? We'll walk through it step by step, so you can ace these types of questions. Understanding how to find the equation of a line using two points is a fundamental concept in coordinate geometry. It enables us to describe the relationship between x and y coordinates along a straight line. This skill is not only crucial for academic purposes but also applicable in various real-world scenarios, such as determining trajectories, designing structures, and analyzing data trends.
Step 1: Understanding the Basics
Before we jump into the solution, let's cover some basics. A straight line can be represented by the equation y = mx + b, where:
- m is the slope (or gradient) of the line.
- b is the y-intercept (the point where the line crosses the y-axis).
The slope (m) tells us how steep the line is. It's calculated as the change in y divided by the change in x (rise over run). The y-intercept (b) is the value of y when x is zero. To find the equation of a line, we need to determine both m and b. Given two points on the line, we can calculate the slope (m) using the formula:
m = (y2 - y1) / (x2 - x1)
where (x1, y1) and (x2, y2) are the coordinates of the two points. Once we have the slope, we can find the y-intercept (b) by substituting one of the points into the equation y = mx + b and solving for b. This involves plugging in the x and y values of the chosen point and the calculated slope into the equation, then isolating b to find its value. With both m and b determined, we can write the complete equation of the line. This method ensures that the line passes through both given points, satisfying the fundamental requirement of the problem.
Step 2: Calculate the Slope (m)
Let's use the points A(2, 3) and B(5, 7) to find the slope. Here’s how we do it:
m = (y2 - y1) / (x2 - x1) = (7 - 3) / (5 - 2) = 4 / 3
So, the slope (m) of the line is 4/3. The slope, often referred to as the gradient, is a measure of the steepness and direction of a line. It quantifies how much the y-value changes for every unit change in the x-value. A positive slope indicates that the line is increasing (going upwards) as you move from left to right, while a negative slope indicates a decreasing line. The magnitude of the slope reflects the steepness of the line; a larger absolute value means a steeper line. In our case, a slope of 4/3 signifies that for every 3 units you move horizontally along the x-axis, the line rises 4 units vertically along the y-axis. This understanding of the slope is crucial for visualizing the line and its behavior on the coordinate plane. Knowing the slope allows us to predict how the y-value will change as the x-value varies, which is essential for many applications in mathematics, physics, and engineering. The calculated slope of 4/3 will now be used to determine the y-intercept (b), completing the information needed to define the line's equation.
Step 3: Find the Y-Intercept (b)
Now that we know the slope, we can plug one of the points into the equation y = mx + b to find the y-intercept. Let's use point A(2, 3):
3 = (4/3) * 2 + b 3 = 8/3 + b b = 3 - 8/3 b = 9/3 - 8/3 b = 1/3
Therefore, the y-intercept (b) is 1/3. The y-intercept, denoted as b, is the point where the line intersects the y-axis. It represents the value of y when x is zero. In the equation y = mx + b, the y-intercept is a constant that shifts the line vertically on the coordinate plane. A positive y-intercept means the line crosses the y-axis above the origin, while a negative y-intercept means it crosses below the origin. In our calculation, the y-intercept of 1/3 indicates that the line intersects the y-axis at the point (0, 1/3). This value is crucial because it, along with the slope, uniquely defines the line. Knowing the y-intercept helps us visualize the line's position on the coordinate plane and understand its behavior near the y-axis. The y-intercept is also useful in various practical applications, such as determining initial conditions in physics problems or fixed costs in economic models. Now that we have both the slope (m = 4/3) and the y-intercept (b = 1/3), we can write the complete equation of the line.
Step 4: Write the Equation of the Line
With m = 4/3 and b = 1/3, the equation of the line is:
y = (4/3)x + 1/3
This is the equation of the line in slope-intercept form. The equation y = (4/3)x + 1/3 represents a line on the coordinate plane where the y value depends on the x value. The slope of 4/3 indicates that for every 3 units you move horizontally, the line rises 4 units vertically. The y-intercept of 1/3 signifies that the line crosses the y-axis at the point (0, 1/3). This equation allows us to find the y value for any given x value on the line. For example, if we plug in x = 0, we get y = 1/3, confirming the y-intercept. If we plug in x = 3, we get *y = (4/3)3 + 1/3 = 4 + 1/3 = 13/3. This equation is a fundamental tool in coordinate geometry and is used to describe and analyze linear relationships. It is also essential for solving systems of linear equations and finding intersections of lines. The slope-intercept form provides a clear and intuitive understanding of the line's properties and behavior.
Step 5: Convert to General Form
To convert the equation to general form, we want to eliminate the fractions and set the equation to zero. The general form is Ax + By + C = 0.
Multiply the entire equation by 3 to get rid of the fractions:
3y = 4x + 1
Rearrange the equation to get the general form:
4x - 3y + 1 = 0
The general form of a linear equation is expressed as Ax + By + C = 0, where A, B, and C are constants, and x and y are variables. This form is particularly useful because it can represent any line, including vertical lines, which cannot be expressed in the slope-intercept form (y = mx + b). Converting the equation to general form involves eliminating fractions and rearranging the terms so that all terms are on one side of the equation, and the equation is set equal to zero. In our case, multiplying the equation y = (4/3)x + 1/3 by 3 gives 3y = 4x + 1. Rearranging this to fit the general form Ax + By + C = 0 results in 4x - 3y + 1 = 0. This form allows for easy comparison of different lines and is often used in solving systems of linear equations. It also provides a standardized way to represent linear equations, making it easier to work with them in various mathematical contexts. The general form is especially convenient when dealing with perpendicular lines, as the slopes of perpendicular lines are negative reciprocals of each other, and this relationship is more easily identified in the general form.
Conclusion
The equation of the line that contains the points A(2, 3) and B(5, 7) in slope-intercept form is y = (4/3)x + 1/3 and in general form is 4x - 3y + 1 = 0. So, none of the initial multiple-choice options were correct. Remember to always double-check your work, guys! Understanding how to derive and manipulate linear equations is crucial for success in mathematics and related fields. By mastering these fundamental concepts, you'll be well-equipped to tackle more complex problems and apply your knowledge in practical applications. Linear equations are the building blocks of many mathematical models and are essential for analyzing and predicting real-world phenomena. So, keep practicing and exploring the world of linear equations! The process we followed involves calculating the slope, finding the y-intercept, and then expressing the equation in both slope-intercept and general forms. This approach is systematic and can be applied to any two given points to find the equation of the line that passes through them.