52 CHAPTER 3. VECTOR PRODUCTS
3.7 Systems Of Equations, GeometryAs you know, equations like 2x+3y = 6 can be graphed as straight lines in R2. To find thesolution to two such equations, you could graph the two straight lines and the ordered pairsidentifying the point (or points) of intersection would give the x and y values of the solutionto the two equations because such an ordered pair satisfies both equations. The followingpicture illustrates what can occur with two equations involving two variables.
x
y
one solution
x
y
two parallel linesno solutions
x
y
infinitelymany solutions
In the first example of the above picture, there is a unique point of intersection. In thesecond, there are no points of intersection. The other thing which can occur is that thetwo lines are really the same line. For example, x+ y = 1 and 2x+ 2y = 2 are relationswhich when graphed yield the same line. In this case there are infinitely many points in thesimultaneous solution of these two equations, every ordered pair which is on the graph ofthe line. It is always this way when considering linear systems of equations. There is eitherno solution, exactly one or infinitely many although the reasons for this are not completelycomprehended by considering a simple picture in two dimensions, R2.
Example 3.7.1 Find the solution to the system x+ y = 3, y− x = 5.
You can verify the solution is (x,y) = (−1,4) . You can see this geometrically by graph-ing the equations of the two lines. If you do so correctly, you should obtain a graph whichlooks something like the following in which the point of intersection represents the solutionof the two equations.
x
(x,y) = (−1,4)
Example 3.7.2 You can also imagine other situations such as the case of three intersectinglines having no common point of intersection or three intersecting lines which do intersectat a single point as illustrated in the following picture.
x
y
x
y
In the case of the first picture above, there would be no solution to the three equationswhose graphs are the given lines. In the case of the second picture there is a solution to thethree equations whose graphs are the given lines.