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Wednesday, November 7, 2012

Two Intersecting Lines

Category: Analytic Geometry, Algebra

"Published in Newark, California, USA"

Find the points of intersection of the following lines:

                          x + 2y - 3 = 0
                         
                          2x - 3y + 8 = 0

Solution:

Since the given equations are all first degree, then they are linear equations. They are straight lines. We can graph the two lines by getting their slope and y-intercept. 

For x + 2y - 3 = 0,

                           x + 2y - 3 = 0
                           2y = -x + 3
                           y = - ½ x + 3/2

                           slope (∆y/∆x), m = - ½
                           y-intercept = 3/2

To trace the graph, plot 3/2 at the y-axis. This is your first point of the line (0, 3/2). Next, use the slope to get the second point. From the first point, count 2 units to the left and then 1 unit upward. 

For 2x - 3y + 8 = 0,

                          2x - 3y + 8 = 0
                          3y = 2x + 8
                          y = ⅔ x + 8/3

                          slope (∆y/∆x), m = 
                          y-intercept = 8/3 or 2 

To trace the graph, plot 8/3 at the y-axis. This is your first point of the line (0, 8/3). Next, use the slope to get the second point. From the first point, count three units to the right and then 2 units upward.


Photo by Math Principles in Everyday Life

From the graph, their point of intersection is (-1, 2). To get their actual point of intersection, we have to use the two given equations and solve for x and y, we have

                                 x + 2y - 3 = 0


                                 2x - 3y + 8 = 0

Multiply the 1st equation by 2 and -1 at the 2nd equation. Add the two equations in order to eliminate x and solve for the value of y.

    2(x + 2y - 3 = 0)             2x + 4y - 6 = 0
                                 
  -1(2x - 3y + 8 = 0)           -2x + 3y - 8 = 0
                                     ________________

                                                7y - 14 = 0
                                                7y = 14
                                                  y = 2

Substitute y to either of the two equations,

                                 x + 2y - 3 = 0
                                 x + 2(2) - 3 = 0
                                 x + 4 - 3 = 0
                                 x + 1 = 0
                                 x = -1

Therefore, their point of intersection is P(-1, 2).


Photo by Math Principles in Everyday Life