Simple Equations

Simple Equations : Simple Equations Session-1

Agenda : Agenda Framing the equations Solving the equations Multiple equations & second order equations

One equation- one unknown : 3x + 5 = 26 ? 3x = 21 ? x = 7 One equation- one unknown

Framing : Twice a number is nine more than half of the number. What is the number? 2x = x/2 + 9 3x/2 = 9 ? x= 6 Framing

Two Equations and two unknowns : Eq1: X + 3Y =10 Eq2: 2X - Y= 13 Eliminate one variable Eq1x 2 2X+6Y = 20 Eq2 2x- Y = 13 7Y = 7 ? Y = 1 X=7 Two Equations and two unknowns

Substitution : The cost of 3 note books and 2 pencils is Rs. 26 and the cost of a note-book is eight times the cost of a pencil. What is the cost of three note-books and three pencils? Framing 3n +2p = 26 n = 8p Substitution 3 (8p) + 2p = 26 26p = 26 ? p=1 From (2) 3n = 26 – 2p = 24 ? n = 8 3n +3p = 3(8) + 3 (1) = 27 Substitution

Example -4 : Father is thrice as old as his son now. Five years ago, he is four times as old as his son. What will be the father’s age after 10 years? F= 3S F-5 = 4 (S-5) Substituting 3S-5 = 4S -20 ? S = 15 S= 15, F= 45 F + 10 = 55 Example -4

Example-5 : Sum of the digits of a number is 10. If the digits are interchanged, the resulting number is 18 more than the original number. What is the original number? Number: xy ? Value: 10x+y x+y =10 --- (1) Interchange Digits: yx Value:10y +x 10y +x – (10x +y) = 18 ? 9y – 9x = 18 ? y-x=2 --- (2) x+y = 10 Solving: y = 6, x=4 Example-5

Example 6 : If the numerator and the denominator of a fraction are increased by 3, the value of the fraction doubles. If the denominator is decreased by 2, the value of the fraction triples. Find the fraction. Original fraction : x/y (x+3)/ (y+3) = 2 (x/y) ---(1) x/ (y-2) = 3 (x/y) ---(2) Eq-1: xy+3y= 2xy+6x ? 3y = xy+6x Eq-2: xy = 3xy-6x ? 2y=6x ? y=3x Substituting in Eq-1, 9x = 3x2+6x ? x=1 Also y=3, Fraction is 1/3 Example 6

Number of solutions? (Unique/ Infinite/ No solution) : 2x+3y = 7 4x+6y = 15 Eq.1 gives 4x+6y = 14 and Eq.2 4x+ 6y =15 Hence there is no solution for this problem. Number of solutions? (Unique/ Infinite/ No solution)

Solve for x and y : 2x+ 3y = 17---(1) 9y = 51- 6x –--(2) Rearrange Eq-2: 6x+9y = 51 ?2x+ 3y =17 Infinite combinations of x and y Hence Infinitely many solutions Solve for x and y

Exceptional cases: Linear equations : 2 pens, 4 pencils and 3 erasers cost Rs.17. One pen and two pencils cost Rs.4. What is the cost of each eraser? 2p + 4P +3e = 17 1p + 2P = 4 Two equations and three unknowns…… 2 ( p+ 2P) + 3e = 17 3e = 9 , or e = 3 Exceptional cases: Linear equations