Question

A system of linear equations with more equations than unknowns is sometimes called an overdetermined system. Can such a system be consistent? Illustrate your answer with a specific system of three equations in two unknowns. Choose the correct answer below. A. Yes, overdetermined systems can be consistent. For example, the system of equations below is consistent because it has the solution nothing. (Type an ordered pair.) x 1 equals 2 comma x 2 equals 4 comma x 1 plus x 2 equals 6 B. No, overdetermined systems cannot be consistent because there are fewer free variables than equations. For example, the system of equations below has no solution. x 1 equals 2 comma x 2 equals 4 comma x 1 plus x 2 equals 12 C. Yes, overdetermined systems can be consistent. For example, the system of equations below is consistent because it has the solution nothing. (Type an ordered pair.) x 1 equals 2 comma x 2 equals 4 comma x 1 plus x 2 equals 8 D. No, overdetermined systems cannot be consistent because there are no free variables. For example, the system of equations below has no solution. x 1 equals 2 comma x 2 equals 4 comma x 1 plus x 2 equals 24

Answer 1

Answer:

A. Yes, overdetermined systems can be consistent.

As, the system of equations below is consistent because it has a solution

$x_{1}=2$, $x_{2}=4$, $x_{1}+x_{2}=6$.

Step-by-step explanation:

We have,

'Over-determined system is a system of linear equations, in which there are more equations than unknowns'.

For e.g. Let us consider the system,

2x - 3y = 1

3x - 2y = 4

x - y = 1

Plotting these equations, we see from the graph below that,

The only intersection point is (2,1). Thus, x= 2 and y= 1 is the solution of this system.

Thus, over-determined system can be consistent.

According to the options,

Option C is not correct as,

$x_{1}=2$, $x_{2}=4$ implies $x_{1}+x_{2}=2+4=6\neq 8$.

Hence, option A is correct.