Equation Of A Line Passing Through A Point With A Given Slope
In the realm of mathematics, understanding the equation of a line is a fundamental concept that serves as a cornerstone for more advanced topics. A line, in its essence, represents a straight path extending infinitely in both directions. To define a line mathematically, we often rely on its slope and a point it passes through. This article delves into the intricacies of determining the equation of a line given a point and its slope, providing a step-by-step guide and exploring the underlying principles.
Understanding Slope-Intercept Form
Slope-intercept form is a powerful tool for representing linear equations. The slope-intercept form equation is expressed as y = mx + b, where m represents the slope of the line and b represents the y-intercept, the point where the line intersects the y-axis. This form provides a clear and concise way to understand the characteristics of a line. The slope (m) quantifies the steepness and direction of the line. A positive slope indicates an upward incline, while a negative slope signifies a downward incline. The magnitude of the slope reflects the rate of change in the y-coordinate for every unit change in the x-coordinate. For instance, a slope of 2 means that for every 1 unit increase in x, the y-value increases by 2 units. Conversely, a slope of -1/2 implies that for every 2 unit increase in x, the y-value decreases by 1 unit. The y-intercept (b) is the point where the line crosses the vertical y-axis. At this point, the x-coordinate is always 0. The y-intercept provides a fixed reference point for the line's position on the coordinate plane. For example, a y-intercept of 3 means that the line intersects the y-axis at the point (0, 3). Slope-intercept form allows us to quickly visualize a line. Knowing the slope and y-intercept, we can easily sketch the line on a graph. Start by plotting the y-intercept (0, b) on the y-axis. Then, use the slope (m) to find another point on the line. Remember, slope is rise over run. So, if the slope is m/1, move up (or down if m is negative) m units from the y-intercept and right 1 unit. Connect these two points to draw the line.
Determining the Equation from a Point and Slope
In many scenarios, we are given a point and the slope of a line and tasked with finding its equation. This is where the slope-intercept form comes into play. The process of finding the equation involves substituting the given point's coordinates and the slope into the slope-intercept form equation y = mx + b and solving for b, the y-intercept. Let's break down the steps involved. First, identify the given information. You will have a point (x₁, y₁) and the slope (m). For example, you might be given the point (2, 5) and a slope of -3. Next, substitute the given values into the slope-intercept form. Replace x with x₁, y with y₁, and m with the given slope in the equation y = mx + b. Continuing the example, this gives you 5 = -3(2) + b. Then, solve for b, the y-intercept. Simplify the equation and isolate b on one side. From the example, 5 = -6 + b. Adding 6 to both sides gives b = 11. Finally, write the equation in slope-intercept form. Now that you have the slope (m) and the y-intercept (b), substitute these values back into the y = mx + b equation. In our example, the equation of the line is y = -3x + 11. To verify the solution, you can check if the given point satisfies the equation. Substitute the x and y coordinates of the original point into the equation you derived. If the equation holds true, your solution is correct. In the example, substituting x = 2 and y = 5 into y = -3x + 11 gives 5 = -3(2) + 11, which simplifies to 5 = -6 + 11, and further to 5 = 5. This confirms that the point (2, 5) does indeed lie on the line y = -3x + 11.
Solving the Specific Problem
Now, let's apply these principles to the specific problem at hand: finding the equation of a line that passes through the point (0, 3) and has a slope of 4. Here, we are given the point (x₁, y₁) = (0, 3) and the slope m = 4. Our goal is to find the equation of the line in the form y = mx + b. Substituting the given values into the slope-intercept form equation, we have: 3 = 4(0) + b. Simplify the equation: 3 = 0 + b. Therefore, b = 3. Now that we have the slope, m = 4, and the y-intercept, b = 3, we can write the equation of the line: y = 4x + 3. This equation represents a line that has a slope of 4 and intersects the y-axis at the point (0, 3). We can verify that the point (0, 3) lies on this line by substituting x = 0 and y = 3 into the equation: 3 = 4(0) + 3, which simplifies to 3 = 3, confirming that the point satisfies the equation. Therefore, the correct equation of the line is y = 4x + 3.
Analyzing the Answer Choices
Now let's consider the provided answer choices:
A. y = -4x + 3 B. y = 4x - 3 C. y = 4x + 3 D. y = -4x - 3
Comparing these options with the equation we derived, y = 4x + 3, we can see that option C matches our solution. Option A has the correct y-intercept but the wrong slope sign (negative instead of positive). Option B has the correct slope but the wrong y-intercept sign (negative instead of positive). Option D has both the wrong slope and the wrong y-intercept signs. Therefore, only option C, y = 4x + 3, correctly represents the equation of the line that passes through the point (0, 3) and has a slope of 4.
Alternative Methods: Point-Slope Form
While the slope-intercept form is a versatile method, another useful tool for finding the equation of a line is the point-slope form. The point-slope form equation is expressed as y - y₁ = m(x - x₁), where m is the slope and (x₁, y₁) is a point on the line. This form is particularly advantageous when you are given a point and the slope but prefer not to solve for the y-intercept directly. Using the point-slope form, we can arrive at the same equation we found using the slope-intercept form. Let's apply the point-slope form to our problem. We have the point (0, 3) and the slope 4. Substituting these values into the point-slope form, we get: y - 3 = 4(x - 0). Simplify the equation: y - 3 = 4x. Add 3 to both sides: y = 4x + 3. As you can see, we arrive at the same equation, y = 4x + 3, using the point-slope form. This demonstrates the flexibility of different approaches in solving mathematical problems. Choosing the method that best suits your understanding and the given information can streamline the problem-solving process.
Real-World Applications of Linear Equations
The concept of linear equations extends far beyond the classroom, finding applications in numerous real-world scenarios. Understanding and manipulating linear equations is crucial for modeling and solving problems in various fields. In physics, for instance, linear equations are used to describe motion with constant velocity. The equation d = vt + d₀, where d is the distance, v is the constant velocity, t is the time, and d₀ is the initial distance, is a linear equation that models the relationship between distance and time for an object moving at a constant speed. In economics, linear equations are used to model supply and demand curves. The demand curve, which shows the relationship between the price of a product and the quantity demanded, is often represented by a linear equation. Similarly, the supply curve, which shows the relationship between the price and the quantity supplied, can also be modeled using a linear equation. These models help economists analyze market trends and make predictions about prices and quantities. Computer graphics relies heavily on linear equations to represent lines and shapes. Lines, the fundamental building blocks of many graphical objects, are defined by linear equations. Transformations such as scaling, rotation, and translation, which are essential for manipulating graphical objects, can be represented using linear transformations. In engineering, linear equations are used in circuit analysis. Kirchhoff's laws, which govern the flow of current and voltage in electrical circuits, are expressed as linear equations. Engineers use these equations to analyze and design circuits, ensuring their proper functioning. Linear equations also play a role in data analysis and statistics. Linear regression, a statistical technique used to model the relationship between two or more variables, involves fitting a linear equation to a set of data points. This technique allows analysts to identify trends and make predictions based on the data. These examples illustrate the widespread applicability of linear equations across diverse fields. A strong understanding of linear equations is therefore an invaluable asset in problem-solving and decision-making in various domains.
Conclusion
In conclusion, finding the equation of a line given a point and its slope is a fundamental skill in mathematics with far-reaching applications. Whether using the slope-intercept form or the point-slope form, the underlying principles remain consistent. By understanding these principles and practicing problem-solving techniques, you can confidently tackle linear equation problems and appreciate their significance in various real-world contexts. The ability to manipulate and interpret linear equations opens doors to deeper understanding and problem-solving across diverse disciplines, from physics and economics to computer graphics and engineering.
