math 416 equations & inequalities ii. solving systems of equations apart from the graphic...
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MATH 416Equations & Inequalities II
Solving Systems of Equations
Apart from the graphic method, there are three other methods we could use to solve equations. These are:
_by Comparison
_by Substitution
_by Elimination through Addition
Solving Systems of Equations
Solving systems of equations by comparison:Example 1, Page 2.2
-4x + 3y = 10
-5x + 8y = 23
Solving Systems of Equations
Solving systems of equations by comparison:
_Isolate same variable in both equations
_Compare equations obtained (one variable)
_Solve variable
_Substitute variable in one equation to obtain second variable
_Test in each original equation
Solving Systems of Equations
Solving systems of equations by comparison:
Practice Ex 2.1, Page 2.6
Solving Systems of Equations
Solving systems of equations by comparison (Special cases):Example 3, Page 2.7
3x + 2y = -5
6x + 4y = 2
Solving Systems of Equations
Solving systems of equations (Special cases):
When both y1 = m1x + n1 & y2 = m2x + n2 expressions have the same slope (m1 = m2), but
different constant term (n1≠ n2), the lines obtained are parallel and the system has
no solution
*Could occur with any of the four methods for solving equations
Solving Systems of Equations
Solving systems of equations by comparison (Special cases):Example 4, Page 2.10
2x + 3y = 7
6x + 9y = 21
Solving Systems of Equations
Solving systems of equations (Special cases):
When both y1 = m1x + n1 & y2 = m2x + n2 expressions have the same slope (m1 = m2), and the
same constant term (n1= n2), the lines obtained are identical and the system has
infinite solutions
*Could occur with any of the four methods for solving equations
Solving Systems of Equations
Solving systems of equations by comparison:
Practice Ex 2.2, Page 2.14
(Only 1, 2, 5, 9, 10)
3, 4 , 6, 7, 8 Homework
Solving Systems of Equations
Solving systems of equations by substitution:Example 1, Page 3.2
7x - 3y = 10
5x -2y = 8
Solving Systems of Equations
Solving systems of equations by substitution:
_Isolate one variable as a function of the other variable in one equation
_Substitute expression obtained in the other equation (results in a one-variable equation)
_Solve variable
_Substitute variable in one equation to obtain second variable
_Test in each original equation
Solving Systems of Equations
Solving systems of equations by substitution:
Practice
Ex 3.1, Page 3.5
Ex 3.2, Page 3.8 (Homework)
Solving Systems of Equations
Solving systems of equations by elimination through addition:Example 3, Page 4.8
4x + y = 19
-3x + 7y = 40
Solving Systems of Equations
Solving systems of equations by elimination through addition:
_Choose one variable to be eliminated
_Transform equations into equivalent to eliminate inverse coefficients of chosen variable
_Add equations
_Solve equation obtained (in one variable)
_Substitute value of variable in one equation to obtain second variable
_Test in each original equation
Solving Systems of Equations
Solving systems of equations by elimination through addition:Example 5, Page 4.11
5x + 4y = 7
3x + 2y =
Solving Systems of Equations
Solving systems of equations by elimination through addition:
Practice
Ex 4.2, Page 4.14