Question

How do you imagine the graph of a three variable system of three equations would look? How about the graph of a system of four equations in four variables?
First, think about a two variable system, the corresponding graph and what each equation in the system represents. Then, expand this idea to three variables and discuss the possible cases for solutions to the system. Finally, extend the idea to four variables.
If you produce three items that each require three inputs, then you can use a system of three equations in three variables to solve. As an example, you could make small , medium and large pizzas ( these would be your variables). Your equations ( constraints) would come from limitations on how much dough , pizza sauce and cheese you have.
How can you relate a point, a line segment, a square and a cube?
How can you relate a point, a line segment , a circle and a sphere? Hint, start with the smallest of these and think about how you could build up to the next one and then the next one.

It is college allgebra.

Answer 1

Geometrically Two variables system with two equation is nothing but the two lines in plane and solution of system is just point of intersection of two lines.

Suppose you have 2 items to eat in breakfast then these are your variables and how much you want to eat these items will create system of equations.

Similarly 3×3 system with 3 variables are nothing but 3 planes in 3D and it's solution is line common to each plane.

Suppose you want to schadule your work of next three days in 3 sessions like morning , afternoon and evening so these will be your variables . Then how much work you want to finish in each session are your coefficients of corresponding variables and for 3 days gets three equations with 3 variables.

Suppose you like to travel in slots of 3 months in year. So these sets of months will give you corresponding equations .like one set will be { January , February, March } and so on . Then 4 variables are sides east, west, north , and south . And how much distance you travel in each direction will give you cofficient of variables and total distance traveled in that slot will be your right hand side of equation . This way you get 4×4 system of equations.

Line segment is collection of points . Then square is formed by Cartesian product of two line segments in 2D and cube is Cartesian product of 3 squares in 3D.

Line segment is collected of points , if we attach two extreme points of segment then we get circle means line segment is homeomorphic to circle and then sphere is obtained by revolving verticle circle around z axis in 3D.

Thank you .