section 3.2 systems of equations in two variables exact solutions by using algebraic computation ...

Section 3.2 Systems of Equations in Two Variables Exact solutions by using algebraic computation The Substitution Method (One Equation into Another) The Elimination Method (Adding Equations) How to identify

Consistent Systems (one solution – lines cross) Inconsistent Systems (no solution – parallel lines) Dependent Systems (infinitely many solutions – same line)

Comparing the Methods

3.2 1

DefinitionSimultaneous Linear EquationsConsider the pair of equations together

4x + y = 10 -2x + 3y = -12

Each line has infinitely many pairs (x, y) that satisfy it.But taken together, only one pair (3, -2) satisfies both.Finding this pair is called solving the system.In 3.1, you learned to solve a system of two equations in

two variables by graphing (approximation). In this section we will learn to solve linear systems

algebraically (precision).3.2 2

Solving Systems of Linear EquationsUsing the Substitution Method

B23

AB

A

B

A

3.2 3

Substitution Method - Example You can pick either variable to start,

you will get the same (x,y) solution. Itmay take some work to isolate a variable:

Solve for (A)’s y or Solve for (A)’s x

443

62

yx

yx

B

A

3.2 4

Solving Systems of Linear EquationsUsing the Elimination (Addition) Method

C

B

A

A21

B

A

3.2 5

Elimination Method – multiply 1 You can pick either equation to multiply.

Sometimes you have to multiply both. Itmay take some work to match up terms:

Multiply A by -2 to eliminate y

1883

2245

yx

yx

B

A

3.2 6

Elimination Method – multiply both When multiplying both equations, pick

the LCD of both coefficients of the samevariable, and insure there are unlike signs:

Eliminate x: Multiply A by 5 and B by -2 (GCD = 10)

2975

1732

yx

yx

B

A

3.2 7

SpecialCases

3.2 8

Inconsistent Systems - how can you tell? An inconsistent system

has no solutions. (parallel lines)Substitution Technique Elimination Technique

23

53

xy

xy

B

A

3.2 9

Dependent Systems – how can you tell? A dependent system has

infinitely many solutions. (same line)

Substitution Technique Elimination Technique

24812

623

xy

xy

B

A

3.2 10

Next Section 3.3 –

Applications: Systems of 2 Equations

3.2 11