> y1 := solve(2*x+3*y=7,y); > y2 := solve(3*x-y=1,y); > x_value := solve(y1=y2,x); > subs(x=x_value,y1); > subs(x=x_value,y2);When asked to find an intersection point, you must remember not to just give the value, but you must also find the corresponding value. Note in the last two Maple lines above, the values were the same as they should be since it is a point of intersection. This same problem can be solved in a much simpler way using Maple's capabilities of solving more than one equation simultaneously. Assuming that we have the same number of unknowns as equations, the syntax for the

> solve({equation1,equation2,equation3,...},{variable1,variable2,variable3,...});The first argument of the

> solve({2*x+3*y=7,3*x-y=1},{x,y});As you can see, this method only required one Maple command to get both the and the coordinates.

Also, recall that not all equations can be solved analytically. When the `solve` command yields a strange looking answer, it does not necessarily mean that there are no solutions. You may have to solve the equations numerically. The `fsolve` command has the same syntax as the `solve` command, however it does not always yield all solutions at once. If your plot indicates that you have more intersection points than what the `fsolve` command has shown, then you must solve for each solution separately using ranges of and values.

Suppose we want to find all intersection points of and . You would find that if you try to use the `solve` command, you would get an answer involving `Root of ...` which means that it cannot be solved analytically. To solve numerically in Maple, you can try the following Maple procedure.

> plot({x^2-4*x-2,3*sin(x)},x=-5..5);The plot shows that there are two intersection points. The commands below show how we can find both.

> fsolve({y=x^2-4*x-2,y=3*sin(x)},{x=-2..0,y=-10..10}); > fsolve({y=x^2-4*x-2,y=3*sin(x)},{x=0..5,y=-10..10});Note the use of the ranges for the and values in the second argument of the

This last example shows a similar procedure using implicit equations. Suppose we want to know where the graph of the equation
intersects with the graph of . First a plot would be necessary to determine the number of intersection points. Then the `fsolve` command can be used with ranges specified for and .

> with(plots): > implicitplot({x*y-y^2+2=2*x^2-1,x/y+y=x*y},x=-5..5,y=-5..5); > fsolve({x*y-y^2+2=2*x^2-1,x/y+y=x*y},{x=-2..0,y=0..1}); > fsolve({x*y-y^2+2=2*x^2-1,x/y+y=x*y},{x=-2..0,y=-1..0});

- Find the intersection point of the two lines and
- By first solving each equation for .
- By solving them as a system of two equations and two unknowns.

- Find all points where the graph of intersects with the graph of .
- Find all points where the graph of the hyperbola intersects with the graph of the circle .

2001-10-30