Download Systems of linear equations

Prof. M Alonso
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.
 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.
Determine if (-1, 6) is a solution of the
system
:
not a solution
•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.
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.
 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
 The
lines never cross, that is, two parallel
lines. This system has NO solution and is
known as the inconsistent system name.
 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.
 Find
the solution
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

2
2
1
 Graph
the second equation –x + y = 1
Y = x + 1
 Y intercept is 1
 Slope is 1
The solution is
the ordered pair
(1, 2)
x y 2
x y 5
 Graph
the first equation
x+y=2
 Graph the second equation x + y = 5
x y 2
x y 5
No solution
x  y 3
2x  2 y  6
Infinite solutions
• 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.
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.
 Solve
2x  y  4
x  y  1
2x  y  4
x  y  1
 Substract
the equations
Thus x = 1
2x  y  4
x  y  1
 Once
we know that x = 1 we substitute
this value in any equation

2( 1) + y = 4

2 +y =4

y =4–2

y=2
 Thus, the solution is (1,2).
x y 2
x y 5
.
Since 0 = -3 is false, that
means there is no solution
x  y 3
2x  2 y  6
 Since
0 = 0 is a true statement this means
that there is an infinite set of ordered pairs
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.
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).