Solutions of linear systems (2.1 old)

Chapter 2Chapter 2Systems of Linear Equations Systems of Linear Equations

and Matricesand Matrices

Section 2.1Section 2.1

Solutions of Linear SystemsSolutions of Linear Systems

Solutions of First - Degree EquationsSolutions of First - Degree Equations

A solution of a first –degree equation A 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 a and the graph of the equation is a straight line.straight line.

Possible SolutionsPossible Solutions Unique solutionUnique solution

Inconsistent SystemInconsistent System

Dependent SystemDependent System

Unique SolutionUnique Solution

When the graphs of two first-degree When the graphs of two first-degree equations intersect, then we say the equations intersect, then we say the point of intersection is the solution of point of intersection is the solution of the system.the system.

This solution is unique in that it is the This solution is unique in that it is the only point that the systems have in only point that the systems have in common.common.

The solution is given by the coordinates The solution is given by the coordinates of the point of intersection.of the point of intersection.

Inconsistent SystemsInconsistent Systems

When the graphs of two first-degree When the graphs of two first-degree equations never intersect (in other equations never intersect (in other words, they are parallel), there is no words, they are parallel), there is no point of intersection.point of intersection.

Since there is not a point that is Since there is not a point that is shared by the equations, then we say shared by the equations, then we say there is no solution.there is no solution.

Dependent SystemDependent System

When the graphs of two first-degree When the graphs of two first-degree equations yield the exact same line, equations yield the exact same line, we say that the equations are we say that the equations are dependent because any solution of dependent because any solution of one equation is also a solution of the one equation is also a solution of the other.other.

Dependent systems have an infinite Dependent systems have an infinite number of solutions.number of solutions.

Solving Systems of EquationsSolving Systems of Equations

There are many methods by which a There 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

Elimination MethodElimination Method

Try to eliminate one of the variables Try to eliminate one of the variables by creating coefficients that are by creating coefficients that are opposites.opposites.

One or both equations may be One or both equations may be multiplied by some value in order to multiplied by some value in order to get opposite coefficients.get opposite coefficients.

Example 1Example 1

Solve the system below and discuss Solve 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

Example 1Example 1

x x + 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 = 123 + 2y = 123 + 2y = 12

2y = 92y = 9 y = 4.5y = 4.5

Example 1Example 1 Check x = 3 and y = 4.5 in other Check 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

Example 2Example 2

Solve the system below and discuss Solve 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

Example 3Example 3

Solve the system below and discuss Solve 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

Example 4Example 4

Solve the system below and discuss Solve 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

Example 5Example 5

Solve the system below and discuss the Solve 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