Equation Of A Line Through (3,3) And (4,5) Y=mx+b Explained
In mathematics, determining the equation of a line is a fundamental concept. Given two points, we can uniquely define a straight line that passes through them. This article will guide you through the process of finding the equation of a line when two points are known. We will use the points (3, 3) and (4, 5) as an example to illustrate the steps involved. Understanding how to find the equation of a line is crucial for various applications in mathematics, physics, engineering, and computer graphics. This skill allows us to model linear relationships, make predictions, and solve real-world problems involving straight lines.
Understanding the Basics
Before we dive into the calculations, let's recap some essential concepts. The equation of a line can be expressed in several forms, but the slope-intercept form is particularly useful for this task. The slope-intercept form is given by:
y = mx + b
where:
- y is the dependent variable (usually plotted on the vertical axis).
- x is the independent variable (usually plotted on the horizontal axis).
- m is the slope of the line, representing the rate of change of y with respect to x.
- b is the y-intercept, the point where the line crosses the y-axis.
To find the equation of a line, we need to determine the values of m and b. Given two points, we can first calculate the slope m and then use one of the points to find the y-intercept b. This approach provides a systematic way to define the line that uniquely passes through the given points.
Calculating the Slope (m)
The slope, often denoted as m, quantifies the steepness and direction of a line. It represents the change in the y-coordinate divided by the change in the x-coordinate between any two points on the line. The formula to calculate the slope (m) given two points (x₁, y₁) and (x₂, y₂) is:
m = (y₂ - y₁) / (x₂ - x₁)
This formula is derived from the concept of "rise over run," where the rise is the vertical change (y₂ - y₁) and the run is the horizontal change (x₂ - x₁). The slope indicates how much the y-value changes for every unit change in the x-value. A positive slope means the line is increasing (going upwards) as you move from left to right, while a negative slope means the line is decreasing (going downwards). A slope of zero indicates a horizontal line, and an undefined slope (division by zero) indicates a vertical line.
Using the points (3, 3) and (4, 5), we can calculate the slope as follows:
m = (5 - 3) / (4 - 3) = 2 / 1 = 2
Thus, the slope of the line passing through these points is 2. This means that for every unit increase in x, the y-value increases by 2. The slope is a critical component of the line's equation, as it determines the line's inclination and direction. A line with a slope of 2 is steeper than a line with a slope of 1, and it rises more quickly as you move along the x-axis.
Finding the Y-Intercept (b)
Once we have the slope, the next step is to find the y-intercept, denoted as b. The y-intercept is the point where the line intersects the y-axis, which occurs when x = 0. To find b, we can use the slope-intercept form of the equation (y = mx + b) and substitute the slope (m) and the coordinates of one of the given points (x, y). This will allow us to solve for b. It is crucial to choose either point correctly to ensure the final equation accurately represents the line passing through both points.
Using the slope we calculated (m = 2) and one of the points, let's use (3, 3), we can substitute these values into the equation:
3 = 2(3) + b
Now, we solve for b:
3 = 6 + b
b = 3 - 6
b = -3
Therefore, the y-intercept (b) is -3. This means the line crosses the y-axis at the point (0, -3). Knowing the y-intercept is essential for defining the line's position on the coordinate plane. The y-intercept, along with the slope, completely specifies the equation of the line in slope-intercept form, making it a fundamental parameter in linear equations.
Constructing the Equation
With the slope (m) and y-intercept (b) determined, we can now construct the equation of the line. We have found that the slope m is 2 and the y-intercept b is -3. Using the slope-intercept form (y = mx + b), we substitute these values to get the equation of the line:
y = 2x - 3
This equation represents the line that passes through the points (3, 3) and (4, 5). The equation uniquely defines the relationship between x and y for all points on the line. To verify the equation, we can substitute the coordinates of the given points into the equation and ensure they satisfy it. For example, for the point (3, 3):
3 = 2(3) - 3
3 = 6 - 3
3 = 3
This confirms that the point (3, 3) lies on the line. Similarly, for the point (4, 5):
5 = 2(4) - 3
5 = 8 - 3
5 = 5
This also confirms that the point (4, 5) lies on the line. Thus, the equation y = 2x - 3 accurately represents the line passing through the given points. The ability to construct such equations is crucial in various mathematical and practical applications, including modeling linear relationships, solving systems of equations, and making predictions based on linear models.
Step-by-Step Solution
To summarize, let's go through the step-by-step solution to find the equation of the line passing through the points (3, 3) and (4, 5):
-
Calculate the slope (m):
Using the formula m = (y₂ - y₁) / (x₂ - x₁):
m = (5 - 3) / (4 - 3) = 2 / 1 = 2
So, the slope m is 2.
-
Find the y-intercept (b):
Using the slope-intercept form y = mx + b and substituting one of the points, let's use (3, 3):
3 = 2(3) + b
3 = 6 + b
b = 3 - 6
b = -3
Thus, the y-intercept b is -3.
-
Construct the equation:
Substitute the values of m and b into the slope-intercept form y = mx + b:
y = 2x - 3
Therefore, the equation of the line passing through the points (3, 3) and (4, 5) is y = 2x - 3. This step-by-step approach ensures accuracy and clarity in finding the equation of a line, and it can be applied to any pair of points in the coordinate plane. Mastering this process is fundamental for more advanced topics in algebra and geometry, as well as for real-world applications involving linear relationships and modeling.
Applications and Importance
Finding the equation of a line passing through two points is not just a theoretical exercise; it has numerous practical applications. In physics, it can be used to describe the motion of an object moving at a constant velocity. In economics, it can model linear cost or revenue functions. In computer graphics, it's used to draw lines and shapes on the screen. Understanding this concept allows us to model and analyze various real-world scenarios using linear equations. Moreover, the ability to determine the equation of a line is a foundational skill for more advanced topics in mathematics, such as calculus and linear algebra. These higher-level concepts build upon the basic understanding of lines and their equations, making this knowledge essential for anyone pursuing further studies in these fields.
Conclusion
In conclusion, finding the equation of a line passing through two points involves calculating the slope, determining the y-intercept, and then constructing the equation using the slope-intercept form. By following the steps outlined in this article, you can confidently find the equation of any line given two points. The equation of the line passing through the points (3, 3) and (4, 5) is y = 2x - 3. This fundamental skill is crucial for various applications in mathematics and other fields, allowing us to model linear relationships and solve problems involving straight lines. Mastering this concept lays a solid foundation for more advanced mathematical studies and practical applications, making it an invaluable skill for students and professionals alike.
