Finding The Equation Of A Line Given Two Points Y=mx+c
Introduction
In the realm of coordinate geometry, a fundamental concept involves determining the equation of a straight line. A straight line is uniquely defined by two points, and various methods exist to derive its equation. This article delves into a step-by-step approach to find the equation of a straight line passing through two given points, expressing the result in the slope-intercept form, y = mx + c, where m represents the slope and c represents the y-intercept. This form is particularly useful as it provides a clear understanding of the line's inclination and where it intersects the y-axis.
The equation of a straight line is a cornerstone of linear algebra and has wide-ranging applications in various fields, including physics, engineering, economics, and computer graphics. Understanding how to derive this equation from two points is a crucial skill for students and professionals alike. This article will not only walk you through the mathematical steps but also provide insights into the underlying concepts, ensuring a comprehensive understanding of the process. We will use a specific example to illustrate the method, making it easier to follow and apply to other similar problems. The goal is to equip you with the knowledge and confidence to tackle such problems with ease and accuracy.
Problem Statement
Let's consider the specific problem of finding the equation of the straight line that passes through the points (5, 4) and (8, 19). Our objective is to express this equation in the form y = mx + c, where m and c are integers or fractions in their simplest forms. This problem serves as an excellent example to demonstrate the method for finding the equation of a line given two points. By working through this example, you will gain a clear understanding of each step involved, from calculating the slope to determining the y-intercept.
This problem is not just a mathematical exercise; it has practical implications. For instance, imagine these points represent data points on a graph, and you want to find a linear model that best fits the data. The equation of the line would then represent this linear model. Understanding how to find this equation is therefore a valuable skill in data analysis and modeling. We will break down the problem into manageable steps, ensuring that you understand the logic behind each step and can apply it to different scenarios. The focus is on clarity and understanding, making this seemingly complex problem accessible to all.
Step 1: Calculate the Slope (m)
The slope of a straight line, often denoted by m, is a measure of its steepness and direction. It quantifies how much the y-coordinate changes for every unit change in the x-coordinate. The formula to calculate the slope given two points, (x₁, y₁) and (x₂, y₂), is given by:
m = (y₂ - y₁) / (x₂ - x₁)
In our case, the given points are (5, 4) and (8, 19). Let's assign (x₁, y₁) = (5, 4) and (x₂, y₂) = (8, 19). Plugging these values into the formula, we get:
m = (19 - 4) / (8 - 5) = 15 / 3 = 5
Therefore, the slope of the line passing through the points (5, 4) and (8, 19) is 5. This means that for every one unit increase in the x-coordinate, the y-coordinate increases by 5 units. The slope is a crucial parameter of the line, as it dictates the line's inclination. A positive slope indicates an upward-sloping line, while a negative slope indicates a downward-sloping line. A slope of 0 represents a horizontal line, and an undefined slope represents a vertical line. Understanding the concept of slope is fundamental to understanding linear equations and their graphical representations.
Step 2: Use the Point-Slope Form
The point-slope form of a linear equation is a convenient way to represent a line when you know its slope (m) and a point (x₁, y₁) that it passes through. The point-slope form is given by:
y - y₁ = m(x - x₁)
We have already calculated the slope, m = 5, and we have two points to choose from: (5, 4) and (8, 19). Let's use the point (5, 4) as (x₁, y₁). Substituting these values into the point-slope form, we get:
y - 4 = 5(x - 5)
This equation represents the line in point-slope form. It is a valid representation of the line, but to express the equation in the desired slope-intercept form (y = mx + c), we need to simplify and rearrange the equation. The point-slope form is particularly useful because it allows us to directly incorporate the slope and a known point on the line. This form is a bridge between the geometric properties of the line (its slope and a point it passes through) and its algebraic representation. By manipulating this equation, we can derive other forms of the linear equation, such as the slope-intercept form, which is the form we are aiming for in this problem.
Step 3: Convert to Slope-Intercept Form (y = mx + c)
The slope-intercept form, y = mx + c, is a widely used representation of a linear equation. It explicitly shows the slope (m) and the y-intercept (c) of the line. The y-intercept is the point where the line crosses the y-axis, and it is the value of y when x is 0. To convert the equation from the point-slope form to the slope-intercept form, we need to simplify and rearrange the equation.
Starting from the point-slope form we derived in the previous step:
y - 4 = 5(x - 5)
First, distribute the 5 on the right side of the equation:
y - 4 = 5x - 25
Next, isolate y by adding 4 to both sides of the equation:
y = 5x - 25 + 4
y = 5x - 21
Now, the equation is in the slope-intercept form, y = mx + c, where m = 5 and c = -21. This form provides a clear picture of the line's behavior. We can immediately see that the line has a slope of 5 and intersects the y-axis at the point (0, -21). The slope-intercept form is particularly useful for graphing linear equations and for comparing different lines. It also allows us to easily determine the y-value for any given x-value, and vice versa. This form is a fundamental tool in linear algebra and has numerous applications in various fields.
Solution
The equation of the straight line that passes through the points (5, 4) and (8, 19) is:
y = 5x - 21
This equation is in the form y = mx + c, where m = 5 and c = -21. The slope, m, is an integer, and the y-intercept, c, is also an integer. This fulfills the requirements of the problem statement, which asked for the equation to be expressed with m and c as integers or fractions in their simplest forms. We have successfully found the equation of the line by following a step-by-step approach, starting from the given points, calculating the slope, using the point-slope form, and finally converting to the slope-intercept form. This process demonstrates a clear and logical method for solving such problems.
The solution y = 5x - 21 represents a unique straight line that passes through the given points. Any other equation would either not pass through both points or would represent a different line altogether. This highlights the fundamental principle that two points uniquely define a straight line. The equation we have found can be used to predict other points on the line, to graph the line, and to analyze its properties. Understanding how to find this equation is therefore a crucial skill in mathematics and its applications.
Conclusion
In conclusion, we have successfully determined the equation of the straight line that passes through the points (5, 4) and (8, 19). We achieved this by following a clear and systematic approach: first, we calculated the slope using the formula m = (y₂ - y₁) / (x₂ - x₁); then, we utilized the point-slope form of a linear equation, y - y₁ = m(x - x₁); and finally, we converted the equation to the slope-intercept form, y = mx + c, which is the desired format. The final equation, y = 5x - 21, provides a complete representation of the line, with a slope of 5 and a y-intercept of -21.
This exercise demonstrates the importance of understanding the different forms of linear equations and how to convert between them. The slope-intercept form is particularly useful for its clarity and ease of interpretation, but the point-slope form is a valuable tool for constructing the equation in the first place. The ability to find the equation of a line given two points is a fundamental skill in mathematics and has applications in various fields. By mastering this skill, you can confidently tackle problems involving linear relationships and gain a deeper understanding of coordinate geometry. The steps outlined in this article provide a clear and concise method for solving such problems, ensuring that you can apply this knowledge to a wide range of scenarios.
