Recall that a line in the coordinate plane is defined by a linear equation relating x and y. That is, if a point (x, y) lies on the line, then those values of x and y satisfy the equation. For instance, the point (3, 2) lies on the line defined by the equation y = 4x − 10, because the equation is true when you plug in x = 3 and y = 2:
y = 4x − 10
2 = 4(3) − 10 = 12 − 10
2 = 2 True
On the other hand, the point (7, 5) does not lie on that line, because the equation is false when you plug in x = 7 and y = 5:
y = 4x − 10
5 = 4(7) − 10 = 28 − 10 = 18? False
So, what does it mean when two lines intersect in the coordinate plane? It means that at the point of intersection, BOTH equations representing the lines are true. That is, the pair of numbers (x, y) that represents the point of intersection solves BOTH equations. Finding this point of intersection is equivalent to solving a system of two linear equations. You can find the intersection by using algebra more easily than by graphing the two lines. Try this example:
At what point does the line represented by y = 4x − 10 intersect the line represented by 2x + 3y = 26?
Because y = 4x − 10, replace y in the second equation with 4x − 10 and solve for x:
2x + 3(4x − 10) = 26
2x + 12x − 30 = 26
14x = 56
x = 4
Now solve for y. You can use either equation, but the first one is more convenient:
y = 4x − 10
y = 4(4) − 10
y = 16 − 10 = 6
Thus, the point of intersection of the two lines is (4, 6).
If two lines in a plane do not intersect, then the lines are parallel. If this is the case, there is no pair of numbers (x, y) that satisfies both equations at the same time.
Two linear equations can represent two lines that intersect at a single point, or they can represent parallel lines that never intersect. There is one other possibility: the two equations might represent the same line. In this case, infinitely many points (x, y) along the line satisfy the two equations (one of which must actually be the other equation in disguise).