Question

(3) (1.1: Geometry) For each part below, give an example of a linear system of
equations in two variables that has the given property. In each case, draw the lines
corresponding to the solutions of the equations in the system.
(a) has no solution
(b) has exactly one solution
(c) has infinitely many solutions
(i) Add or remove equations in (b) to make an inconsistent system.
(ii) Add or remove equations in (b) to create infinitely many solutions.
(iii) Add or remove equations in (b) so that the solution space remains unchanged.
(iv) Can you add or remove equations in (b) to change the unique solution you had
to a different unique solution?
In each of (i) - (iv) justify your action in words.

Help with B and i - iv especially please

Answer 1

(a) The system of linear equations has no solution

X+Y=1, X+Y=0

(b)The system of linear equations has single solution

X+Y=1, X-Y=0

(c) The system of linear equations has infinite solution

X+Y=1, 2X+2Y=2

(i) A system of linear equations with no solutions is called an inconsistent system.

let consider the (b) X+Y=1, X-Y=0

Adding 2Y in equation X-Y =0 will convert equation 2 into X+Y=0

Now the new equation sets X+Y=1 and X+Y=0 is inconsistent

(ii) let consider the (b) X+Y=1, X-Y=0

Adding X+3Y=2 in equation X-Y=0 will convert equation 2 into 2X+2Y=2

Now the new equation sets X+Y=1 and 2X+2Y=2 has many solutions

(iii) let consider the (b) X+Y=1, X-Y=0

Adding X-Y=0 in equation 2 will convert it to 2X-2Y=0

Now the new equation sets X+Y=1, 2X-2Y=0 the solution space remains unchanged

(iv) let consider the (b) X+Y=1, X-Y=0

Adding X to the equation 2 will convert it to 2X-Y=0

Now the new equation sets X+Y=0, 2X-Y=0 has unique solution different from the first solution