MATH 416 Equations & 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 x + 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 y 1 = m 1 x + n 1 & y 2 = m 2 x + n 2 expressions have the same slope (m 1 = m 2 ), but different constant term (n 1 ≠ n 2 ), 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 x + 3y = 7 6x + 9y = 21

Solving Systems of Equations Solving systems of equations (Special cases): When both y 1 = m 1 x + n 1 & y 2 = m 2 x + n 2 expressions have the same slope (m 1 = m 2 ), and the same constant term (n 1 = n 2 ), 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: Practice Ex 4.2, Page 4.14