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