1. Chapter 2Chapter 2
Systems of Linear EquationsSystems of Linear Equations
and Matricesand Matrices
Section 2.1Section 2.1
Solutions of Linear SystemsSolutions of Linear Systems

2. Solutions of First - Degree EquationsSolutions of First - Degree Equations
 A solution of a first –degree equationA solution of a first –degree equation
in two unknowns is an ordered pair,in two unknowns is an ordered pair,
and the graph of the equation is aand the graph of the equation is a
straight line.straight line.

3. Possible SolutionsPossible Solutions
 Unique solutionUnique solution
 Inconsistent SystemInconsistent System
 Dependent SystemDependent System

4. Unique SolutionUnique Solution
 When the graphs of two first-degreeWhen the graphs of two first-degree
equations intersect, then we say theequations intersect, then we say the
point of intersection is the solution ofpoint of intersection is the solution of
the system.the system.
 This solution is unique in that it is theThis solution is unique in that it is the
only point that the systems have inonly point that the systems have in
common.common.
 The solution is given by the coordinatesThe solution is given by the coordinates
of the point of intersection.of the point of intersection.

5. Inconsistent SystemsInconsistent Systems
 When the graphs of two first-degreeWhen the graphs of two first-degree
equations never intersect (in otherequations never intersect (in other
words, they are parallel), there is nowords, they are parallel), there is no
point of intersection.point of intersection.
 Since there is not a point that isSince there is not a point that is
shared by the equations, then weshared by the equations, then we
say there is no solution.say there is no solution.

6. Dependent SystemDependent System
 When the graphs of two first-degreeWhen the graphs of two first-degree
equations yield the exact same line,equations yield the exact same line,
we say that the equations arewe say that the equations are
dependent because any solution ofdependent because any solution of
one equation is also a solution of theone equation is also a solution of the
other.other.
 Dependent systems have an infiniteDependent systems have an infinite
number of solutions.number of solutions.

7. Solving Systems of EquationsSolving Systems of Equations
 There are many methods by which aThere are many methods by which a
system of equation can be solved:system of equation can be solved:
• GraphingGraphing
• Echelon Method (using transformations)Echelon Method (using transformations)
• Substitution MethodSubstitution Method
• Elimination MethodElimination Method

8. Elimination MethodElimination Method
 Try to eliminate one of the variablesTry to eliminate one of the variables
by creating coefficients that areby creating coefficients that are
opposites.opposites.
 One or both equations may beOne or both equations may be
multiplied by some value in order tomultiplied by some value in order to
get opposite coefficients.get opposite coefficients.

9. Example 1Example 1
 Solve the system below and discussSolve the system below and discuss
the type of system and solution.the type of system and solution.
x + 2y = 12x + 2y = 12
-3x – 2y = -18-3x – 2y = -18

10. Example 1Example 1
xx + 2y+ 2y = 12= 12
-3x-3x – 2y– 2y = -18= -18
-2x = -6-2x = -6
x = 3x = 3
Solve for y:Solve for y: x + 2y = 12x + 2y = 12
3 + 2y = 123 + 2y = 12
2y = 92y = 9
y = 4.5y = 4.5

11. Example 1Example 1
 Check x = 3 and y = 4.5 in otherCheck x = 3 and y = 4.5 in other
equation.equation.
-3x – 2y = -18-3x – 2y = -18
-3(3) – 2(4.5) = -18-3(3) – 2(4.5) = -18
-9 – 9 = -18-9 – 9 = -18
-18 = -18 √-18 = -18 √
 Solution:Solution:
Unique solution: (3, 4.5)Unique solution: (3, 4.5)
Independent systemIndependent system

12. Example 2Example 2
 Solve the system below and discussSolve the system below and discuss
the type of system and solution.the type of system and solution.
2x – y = 32x – y = 3
6x – 3y = 96x – 3y = 9

13. Example 3Example 3
 Solve the system below and discussSolve the system below and discuss
the type of system and solution.the type of system and solution.
x + 3y = 4x + 3y = 4
-2x – 6y = 3-2x – 6y = 3

14. Example 4Example 4
 Solve the system below and discussSolve the system below and discuss
the type of system and solution.the type of system and solution.
4x + 3y = 14x + 3y = 1
3x + 2y = 23x + 2y = 2

15. Example 5Example 5
 Solve the system below and discuss theSolve the system below and discuss the
type of system and solution.type of system and solution.
x 6+ y=
5 5
yx 5+ =
10 3 6