Question

10.) Consider the linear equation 2a + 3b = 4

Is (a; b) = ( 12 ; 1) a solution to the equation? Motivate your answer.

11.) Look up what is meant by a system of linear equations.

A known fact of solutions of systems of linear equations is that only one the following options can hold :

(a) No solution possible

(b) A unique solution can be found

(c) The system has infinite solutions.

Consider that two straight lines form a linear system.

Interpret what happens geometrically to the straight lines to get each case of the solution types given above.

12.) Look up the concept of a homogeneous linear system.

Only two solution types of the three mentioned solution types above are possible. Which one can never happen and why.

Answer 1

10.) Consider the linear equation

$$2a + 3b = 4$$

If $(a, b)=(\dfrac{1}{2}, 1),$ then substitute

$$2(\dfrac{1}{2}) + 3(1) = 4$$

$$4=4, True$$

Therefore $(a, b)=(\dfrac{1}{2}, 1)$ is a solution to the equation $2a+3b=1.$

11)

a) Two lines are parallel lines or skew lines.

b) Two lines are intersecting lines.

c) Two lines are coincident lines.

12) Since a homogeneous system always has a solution (the trivial solution), it can never be inconsistent.

Therefore a homogeneous linear system can have:

(a) A unique solution.

Or

(b) Infinite solutions.