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SYSTEMS OF LINEAR EQUATIONS
Definitions. A system of two linear equations consists of two equations of degree one in the
same two variables.
Example:
x+y=6
3x - y = 2
A solution of a system of two linear equations in two variables is an ordered pair of numbers
satisfying both equations simultaneously.
In the above example (2,4) is a solution of both equations. In fact, 2+4 = 6 and 3(2)-4 = 2.
Therefore the ordered pair (2,4) is a solution of the system.
Graphical Solutions. One way of solving a system of two equations in two unknowns
consists of graphing both equations in the same Cartesian plane. If both lines coincide
(overlap) the system id said to be indeterminate, and the solution is the infinite set of pairs of
numbers satisfying any of the two equations. If the lines are parallel the system is said to be
incompatible, and it has no solution. Finally, if the two lines intersect each other the system
is called determinate, and the solution is the ordered pair of numbers corresponding to the
intersection point.
Example: Using the equations from the previous example, we have
The solution (2,4) corresponds to the intersection point of both lines.
Algebraic Methods. In many instances solving systems of equations graphically is not the
best way of getting a satisfactory answer. Such cases include those in which the coordinates
of the intersection point are not integers and therefore cannot be read accurately.
Instead of estimating the solution we can use one of two algebraic techniques that will be
discussed in this lesson: The Elimination by Addition and Substitution methods.
Elimination by Addition. This technique is based on the Addition Principle of Equality.
According to this Principle if equal amounts are added to both sides of an equation its
solution remains unchanged.
Example: Given the system
3x + y = 4
2x - y = 11
If we add both equation together we get
5x = 15
From which
x=3
Substituting this value of x into one of the original equations and solving for y we get
3(3) + y = 4
y = -5
The solution of the system is then the ordered pair (3,-5).
Notice that the combined equation 5x = 15 has only one unknown. This is because in the
given equations the coefficients of y are opposite numbers.
Many times this does not occur and the Multiplication Principle of Equality must be applied
to one or both equations. This Principle says that the solutions of an equation do not change
if we multiply both sides of the equation by the same non-zero number.
Example: Solve the system
3x + 2y = 12
5x - 3y = 1
Multiplying the first equation by 3 and the second by 2 we get:
9x + 6y = 36
10x - 6y = 2
Adding both equations together
19x = 38
Solving for x
x=2
Substituting x =2 into one of the given equations (let's say the first)
3(2) + 2y = 12
Solving for y
2y = 6
y=3
The solution of the system is the ordered pair (2,3). In fact,
3(2) + 2(3) = 12
and
5(2) - 3(3) =1.
Substitution Method. To solve a system of linear equations by substitution we proceed as
follows:
First, we solve one of the given equations for one of the variables.
Second, we substitute the resulting expression into the other equation.
Third, we solve the combined equation.
Fourth, we substitute the solution from the third step into one of the original equations.
Fifth, we solve the resulting equation.
Example: Solve the following system by substitution
x - y =6
3x + y = 2
Solving the first equation for x gives
x=y+6
Substituting this value into the second equation
3(y + 6) + y = 2
Solving for y
3y + 18 + y = 2
4y = -16
y = -4
Substituting this value of y in the equation x = y + 6 we get
x = -4 + 6 or x = 2
The solution of the system is the ordered pair (2,-4). In fact, 2 - (-4) = 6, and 3(2) + (-4) = 2.
Remarks. i. If one or both equations of a system contain denominators and/or grouping
symbols, we reduce the equation to general form (ax + by = c) before applying any method
of resolution.
Example: Solve the system
x y x y

2
3
2
x 2x  y

 10
3
7
-Multiplying the first equation by 6
-Removing parentheses
-Combining like terms
-Multiplying the second equation by 21
-Removing parentheses
-Combining like terms
:
:
:
:
:
:
2(x + y) - 3(x - y) = 12
2x + 2y - 3x + 3y = 12
- x + 5y = 12
7x - 3(2x + y) = 210
7x - 6x - 3y = 210
x - 3y = 210
-The system is then reduced to:
- x + 5y = 12
x - 3y = 210
-Adding both equation together:
2y = 222
-Solving for y:
y = 111
-Substituting in the second equation:
x - 3(111) = 210
-Solving for x:
x = 543
Therefore the solution of the system is (543,111).
In fact, substituting these values into the given system we have
(543 + 111)/3 - (543 - 111)/2 = 218 - 216 = 2, and
543/3 - (2(543) + 111)/7 = 181 - 171 = 10.
ii. The Elimination and Substitution methods can be easily extended and combine to solve
systems of three or more equations in three or more variables.
Example:
Solve the system
x+y+z=6
x+y-z=4
2x - y + z = 5
-Adding the last two equations:
3x = 9
-Solving for x
x=3
-Substituting x=3 into the first two equations
3 + y +z = 6
3+y-z=4
-Simplifying
y+z=3
y-z=1
-Adding these two equations together
2y = 4
-Solving for y
y=2
-Substituting y=2 into the first of the above two equations
2+z=3
-Solving for z
z =1
The solution of the system is the ordered triple (3,2,1).
In fact, substituting these values into the given equations we get the identities
3+2+1=6
3+2-1=4
2(3) - 2 + 1 = 5
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EXERCISES
1. Solve graphically the following system
x + 3y = 9
2x - y = 4
2. Solve by elimination
8x + 5y = 3
3x + 2y = -1
3. Solve by substitution
x - 4y = 18
-6x + 3y = -3
4. Solve by any method
5x  4 x  5 y 2


4
2
3
x  2y  8
5. Solve the system
2x + y +z = 4
6x - y - z = 0
4x - 3y +2z = -7
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