Slide 1:ALGEBRA Engr. Lizette Ivy G. Catadman Instructor / Lecturer LECTURE NOTES & PROBLEM SETS: PART 5


LINEAR EQUATIONS :LINEAR EQUATIONS EQUATION IT IS A STATEMENT THAT TWO EXPRESSIONS ARE EQUAL. THE EXPRESSIONS ARE CALLED MEMBERS OR SIDES OF THE EQUATION. LINEAR EQUATION IT IS AN EQUATION OF THE FIRST DEGREE.


Slide 3:IDENTICAL EQUATION OR IDENTITY IT IS AN EQUATION WHOSE MEMBERS ARE EQUAL FOR ALL ADMISSIBLE VALUES OF THE UNKNOWN WHICH IT CONTAINS. CONDITIONAL EQUATION IT IS AN EQUATION WHOSE MEMBERS ARE EQUAL FOR CERTAIN VALUES (OR FOR NO VALUES) OF THE UNKNOWN/S WHICH IT CONTAINS, BUT NOT FOR ALL ADMISSIBLE VALUES.


Slide 4:ROOT SOLUTION OF THE EQUATION IT IS THE NUMBER/S WHICH, WHEN SUBSTITUTED FOR THE UNKNOWNS, MAKES THE MEMBERS OF THE EQUATION EQUAL TO EACH OTHER. TO CHECK, SUBSTITUTE THE OBTAINED VALUE OF X INTO THE ORIGINAL EQUATION.


Slide 5:EQUIVALENT EQUATIONS EQUATIONS HAVING EXACTLY THE SAME SOLUTION. BOTH EQUATIONS HAVE THE SAME SOLUTION OR ANSWER, x = 7. OPERATIONS THAT LEAD TO AN EQUIVALENT EQUATION: ADDING OR SUBTRACTING THE SAME NUMBER OR EXPRESSION TO BOTH SIDES OF THE EQUATION. MULTIPLYING OR DIVIDING BOTH SIDES BY THE SAME NUMBER OR EXPRESSION, EXCEPT ZERO AND DOES NOT CONTAIN AN UNKNOWN.


SOLUTION OF A LINEAR EQUATION WITH ONE UNKNOWN :SOLUTION OF A LINEAR EQUATION WITH ONE UNKNOWN IF FRACTIONS APPEAR, PLACE PARENTHESES AROUND EACH NUMERATOR AND CLEAR OFF FRACTIONS BY MULTIPLYING BOTH MEMBERS BY THE LCD OF THE FRACTIONS. REMOVE THE PARENTHESES BY COMBINING LIKE TERMS. TRANSPOSE ALL TERMS CONTAINING THE UNKNOWN ON ONE SIDE AND ALL THE OTHER TERMS ON THE OTHER SIDE. DIVIDE BOTH SIDES BY THE COEFFICIENT OF THE UNKNOWN.


SOLUTION OF A LINEAR EQUATION WITH ONE UNKNOWN :SOLUTION OF A LINEAR EQUATION WITH ONE UNKNOWN CONTINUED... CHECK BY SUBSTITUTING THE RESULT INTO THE ORIGINAL EQUATION.


SOLUTION OF A LINEAR EQUATION WITH TWO UNKNOWNS :SOLUTION OF A LINEAR EQUATION WITH TWO UNKNOWNS GRAPHICAL SOLUTION OF SYSTEM OF TWO EQUATIONS A SOLUTION OF A SYSTEM OF TWO EQUATIONS WITH TWO UNKNOWNS (x AND y) IS A PAIR OF VALUES (x, y) WHICH SATIFY BOTH EQUATIONS. IF A SYSTEM HAS A SOLUTION, THE EQUATIONS ARE CALLED SIMULTANEOUS. A SYSTEM OF TWO LINEAR EQUATIONS WITH TWO UNKNOWNS USUALLY HAS JUST ONE [PAIR] SOLUTION BUT CERTAIN CASES MAY OCCUR.


SOLUTION OF A LINEAR EQUATION WITH TWO UNKNOWNS :SOLUTION OF A LINEAR EQUATION WITH TWO UNKNOWNS GRAPHICAL SOLUTION OF SYSTEM OF TWO EQUATIONS (CONTINUED...) IF THE GRAPH OF THE EQUATIONS ARE PARALLEL LINES, THE SYSTEM HAS NO SOLUTION AND THE EQUATIONS ARE CALLED INCONSISTENT EQUATIONS. IF THE GRAPHS ARE ON THE SAME LINE, THE SOLUTION OF ONE IS ALSO A SOLUTION OF THE OTHER. THE SYSTEM HAS INFINITE SOLUTTIONS. THEY ARE SAID TO BE DEPENDENT EQUATIONS.


SOLUTION OF A LINEAR EQUATION WITH TWO UNKNOWNS :SOLUTION OF A LINEAR EQUATION WITH TWO UNKNOWNS ANALYTICAL SOLUTION OF SYSTEM OF TWO, THREE, FOUR, ... EQUATIONS ELIMINATION BY ADDITION OR SUBTRACTION ELIMINATION BY SUBSTITUTION CRAMER’S RULE EMPLOYING DETERMINANTS (FOR N3, DIAGONAL AND FOR N3, CO-FACTOR METHOD)


Slide 11:USING ELIMINATION BY ADDITION OR SUBTRACTION: MULTIPLY EQ. 1 BY 2, RETAIN EQ. 2. USING ELIMINATION BY ADDITION OR SUBTRACTION: MULTIPLY EQ. 1 BY 5, MULTIPLY EQ. 2 BY 3.


Slide 12:USING ELIMINATION BY SUBSTITUTION: OBTAIN x = THE REST OF THE EQUATION FROM EQ. 1. SUBSTITUTE IT TO EQ. 2. SUBSTITUTE THE VALUE OF y INTO THE x.


Slide 13:USING CRAMER’S RULE (EMPLOYING DETERMINANTS)


Slide 14:USING ELIMINATION BY ADDITION OR SUBTRACTION TAKE EQ. 1 AND EQ. 2. ELIMINATE UNKNOWN “B”. TAKE EQ. 2 AND EQ. 3. ELIMINATE UNKNOWN “B”. EQUATION 4 EQUATION 5


Slide 15:TAKE EQ. 4 AND EQ. 5. ELIMINATE UNKNOWN “A”. SUBSTITUTE “C” IN EQ. 4. SUBSTITUTE “A” AND “C” IN EQ. 1.


Slide 16:USING ELIMINATION BY SUBSTITUTION TAKE EQ. 1, OBTAIN A = TO THE REST OF THE EQUATION. SUBSTITUTE IN EQ. 2 AND EQ. 3.


Slide 17:USING CRAMER’S RULE EMPLOYING DETERMINANTS


Slide 18:WHEN THE NUMBER OF EQUATIONS IS EQUAL TO OR GREATER THAN 3, CRAMER’S RULE EMPLOYING DETERMINANTS IS USED BUT THE SOLUTION FOR THE DETERMINANTS MUST BE DONE USING CO-FACTOR METHOD. DIAGONAL METHOD IS NO LONGER VALID. USING CRAMER’S RULE EMPLOYING DETERMINANTS