Stelsels lineaire vergelijkingen *: Twee vergelijkingen met twee onbekenden *
Equations and lines
Two line in the flat plane can be in three different positions relative to each other:
- they can cross (or: intersect in one point),
- they can be parallel but unequal, or
- they can be identical.
The equations of these lines with unknowns #x# and #y# have, in the three respective cases: one solution (regular), no solution (contradicting), and the same line as solution (dependent).
In all cases, the points which lie on both lines are the solution to the system of two equations.
The first case occurs then and only then if the slopes of these two lines are different.
The statements follow from the fact that the following three statements are equivalent:
- the point #[a,b]# lies on the line #cx+dy=f#
- #x=a\land y=b# is a solution to the equation #cx+dy=f#
- #c\cdot a +d\cdot b=f#
Regular: two intersecting lines. Define the intersection point of the equations #y=4x−1# and #y=2x+3#.
At the intersection point, the #y#-values of the point are equal on both lines. This gives the linear equation #4x−1=2x+3#. The solution to this equation is #x=2#. This is the #x#-coordinate of the intersection point of these two lines. The #y#-coordinate can be found by entering the value of #x# in in one of the two equations: #y=4x−1=4⋅2−1=7#. Hence, the solution to the equation is #x=2\land y=7# (which can also be written as #[x,y]=[2,7]#) and the intersection point of the two lines is #[2,7]#.
Contradicting: two parallel lines. Define the intersection point of the two lines with equations #y=3−x# and #x+y=1#.
In the equation #x+y=1# replace the unknown #y# by #3−x#, and reduce the result #x+(3−x)=1#. This gives #2 = 0#, a contradiction. There is no solution and no intersection point. In other words: the two lines are parallel.
Dependent: two overlapping lines. Define the intersection point of the two lines with equations #y=−x+2# and #2x+2y=4#.
In the equation #2x+2y=4# replace the unknown #y# by #−x+2#. This gives #2x+2(−x+2)=4#, which can be reduced to #0=0#. Apparently each #x# is a solution to this equation. The with #x# corresponding #y#-coordinate #y=-x+2# satisfies both equations. Hence, the solutions can be described by one of the unknowns, say #x#, as a parameter and expressing the value of #y# in that parameter: #y=2-x#. All points of the one line, are on the other. In other words: the lines overlap. Or: the two equations describe the same line.
This can also be seen by moving all terms in both equations to the left hand side, and dividing all terms in the second equation by #2#. Because the emerging equations are equal, the two original equations are said to be equivalent.
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