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H. 1994. Elementary Linear Algebra. 7th Edition. John Wiley & Sons Inc. 2.
Systems of Linear Equations Solution Sets Aim To introduce and explain the possible solution sets associated with systems of linear equations. Learning Outcomes At the end of this section you will be able to: • Tell the difference between a consistent and an inconsistent system of equations, • Understand the different solution sets.
Not all systems of linear equations have solutions. Take for example the following system, x+y = 4 2x + 2y = 6 If we multiply the second equation of the system by
1 2
it is clear to see that there is no
solution since the resulting system x+y = 4 x+y = 3 has contradictory equations. A system of equations that has no solution is said to be inconsistent; if there is at least one solution of the system, it is said to be consistent. To demonstrate the possibilities that can occur in solving systems of linear equations, consider a general system of two linear equations in the unknowns x and y: a1 x + b1 y = c1
(a1 , b1 not both zero)
a2 x + b2 y = c2
(a2 , b2 not both zero)
The graphs of these equations are lines, call them l1 and l2 . Since a point (x, y) lies on a line if and only if the numbers x and y satisfy the equation of the line, the solutions of the system of equations correspond to points of intersection of l1 and l2 . 1
Systems of Linear Equations There are three possibilities (show in the figure below): • The lines l1 and l2 may be parallel, in which case there is no intersection and consequently no solution of the system. • The lines l1 and l2 may intersect at only one point, in which case the system has exactly one solution. • The lines l1 and l2 may coincide, in which case there are infinitely many points of intersection and consequently infinitely many solutions to the system. These are the only possibilities when dealing with linear systems. Note: Every system of linear equations has either no solutions, exactly one solution, or infinitely many solutions. l1
l1
l2
l1 and l2
l2
(a)
(b)
(c)
• (a) - No Solution • (b) - One Solution • (c) - Infinitely Many Solutions
Related Reading Anton, H. 1994. Elementary Linear Algebra. 7th Edition. John Wiley & Sons Inc.
