Systems of linear equations Prof. M Alonso

Definitions A system of linear equations consists of two or more linear equations. The solution of a system of linear equations in two variables is any ordered pair that solves both linear equations.

Examples

Example Determine whether the point (1, 2) is a solution of the following system. Since the point (1, 2) produces a true statement in both equations, it is a solution.

Example Determine if (-1, 6) is a solution of the system : not a solution

Solving a system by graphing A solution of a system of equations is a solution common to both equations, thus it would also be a point common to the graphs of both equations. To find the solution of a system of 2 linear equations, graph the equations and see where the lines intersect.

Graphing Method The graphical method consists of drawing the graph of both equations. Remember that the graph of these equations are straight lines. Therefore, visually we will have two lines in the Cartesian plane. If we have two lines in the plane, three possibilities can occur.

lines intersect The lines intersect at one point. The point where they intersect is the solution of the system. This system is known by the name of independent system

lines never cross The lines never cross, that is, two parallel lines. This system has NO solution and is known as the inconsistent system name.

One line on top of the other One line is on top of the other and therefore we see only one straight line. This system has infinite solutions and is known by the name of dependent system.

Example Find the solution

Solve First graph 2x + y = 4. Y = -2x + 4 The y intercept is (0,4) and the slope is -2. Remember that the slope can be written as

First line 2x + y = 4

Second equation –x + y = 1 Graph the second equation –x + y = 1 Y = x + 1 Y intercept is 1 Slope is 1

Graph of the system The solution is the ordered pair (1, 2)

Graph Graph the first equation x + y = 2 Graph the second equation x + y = 5

Graph of No solution

Solve Infinite solutions

Summary There are three possible outcomes when graphing two linear equations in a plane. One point of intersection, so one solution Parallel lines, so no solution Coincident lines, so infinite number of solutions If there is at least one solution, the system is considered to be consistent. If the system defines distinct lines, the equations are independent.

Eliminitation Method Another method that can be used to solve systems of equations is called the addition or elimination method. Multiply both equations by numbers that will allow you to combine the two equations and eliminate one of the variables.

EXAMPLE Solve

Example Substract the equations Thus x = 1

Solve Once we know that x = 1 we substitute this value in any equation Thus, the solution is (1,2).

Solve . Since 0 = -3 is false, that means there is no solution

Solve Since 0 = 0 is a true statement this means that there is an infinite set of ordered pairs

Summary Solving a System of Linear Equations by the Elimination Method Rewrite each equation in standard form, eliminating fraction coefficients. If necessary, multiply one or both equations by a number so that the coefficients of a chosen variable are opposites. Add the equations Find the value of one variable Find the value of the second variable by substituting the value found in one equation.

Summary Use of the addition method to combine two equations might lead you to results like 0 = 0 (which is always true, thus indicating that there are infinitely many solutions, since the two equations represent the same line) 8 = 6 (which is never true, thus indicating that there are no solutions, since the two equations represent parallel lines).