Determining the Equation of a Line Passing Through Given Points
In the field of mathematics, particularly in algebra, finding the equation of a line that passes through specified points is a fundamental concept. This process is crucial for various applications, from graphing to real-world scenarios. This article will guide you through the steps of determining the equation of a line passing through the points (1, 2) and (5, 4).
Introduction
Given two points on a line, you can determine the equation of that line. This involves solving a system of linear equations. Let's explore the process in detail.
Setting Up the Equations
Given the points (1, 2) and (5, 4), we can set up the following equations:
For the point (1, 2): (1a b 2) For the point (5, 4): (5a b 4)These equations are based on the general form of a linear equation, (ax by c), where (a) and (b) are the coefficients we need to determine.
Solving the System of Equations
The next step is to solve the system of equations to find the values of (a) and (b).
Substitution Method
From the first equation, we can solve for (a) in terms of (b): [a 2 - b]
Now, substitute this expression for (a) into the second equation:
[5(2 - b) b 4]
Simplify and solve for (b): [10 - 5b b 4]
[-4b -6]
[b frac{3}{2}]
Solving for (a)
Now that we have the value of (b), we can substitute it back into the expression for (a): [a 2 - frac{3}{2} frac{4}{2} - frac{3}{2} frac{1}{2}]
Forming the Equation of the Line
With the values of (a) and (b) determined, we can form the equation of the line. The general form is (y ax b). Substituting our values, we get:
[y frac{1}{2}x frac{3}{2}]
Visualization and Verification
For a clearer understanding, you can plot the points (1, 2) and (5, 4) on a graph and verify that the line (y frac{1}{2}x frac{3}{2}) indeed passes through these points. This can be done using graphing software or an online graphing tool.
Conclusion
Determining the equation of a line passing through given points involves setting up a system of equations and solving for the coefficients. By following the steps outlined in this article, you can confidently find the equation of any line given two points. This skill is not only useful in mathematics but also has applications in fields such as physics, engineering, and data analysis.