Finding the Intersection Point of Two Lines Using Algebra and Graphing
In this article, we will explore a practical problem that involves finding the coordinates of the point where two lines intersect. The problem is as follows: Consider the equations of two lines 2xy - 4 0 and 3x - y - 1 0. What are the coordinates of the point where these lines intersect?
Algebraic Method for Solving Linear Equations
To solve this problem algebraically, we first need to rewrite the given equations in the form y mx b where m is the slope and b is the y-intercept.
The first equation is 2xy - 4 0. Rearranging this equation, we get:
y -2x 4
The second equation is 3x - y - 1 0. Rearranging this equation, we get:
y 3x - 1
Now, we set the expressions for y equal to each other to find the x-coordinate of the intersection:
-2x 4 3x - 1
-5x -5
x 1
Substitute x 1 into either of the original equations to solve for y. Using the first equation:
2(1) - y - 4 0
2 - y - 4 0
-y 2
y 2
Therefore, the coordinates of the point where the two lines intersect are (1, 2).
Graphical Method for Solving Linear Equations
Alternatively, we can solve for the intersection point graphically. Let's write the equations as:
Line 1: 2xy - 4 0 can be rewritten as y -2x 4
Line 2: 3x - y - 1 0 can be rewritten as y 3x - 1
If we plot these equations on a graph, the point of intersection is where the two lines cross. We can see that the intersection point is (1, 2).
Alternative Algebraic Approach
Another way to solve this problem is by adding the two equations directly:
2xy - 4 0
3x - y - 1 0
When we add the above two equations, we get:
5x - 5 0
x 1
Substitute x 1 into either of the original equations to find y. Using the first equation:
2(1) - y - 4 0
2 - y - 4 0
-y 2
y 2
Therefore, the coordinates of the intersection point are (1, 2).
Conclusion
The intersection point of the lines given by the equations 2xy - 4 0 and 3x - y - 1 0 is (1, 2). This problem demonstrates the importance of understanding both algebraic and graphical methods for solving such problems.