Linear Equations

Linear EquationsIn One VariableDefinationA Linear equation is an equation involving single variables or literals which have highest power 1, i.e. we have linear equation as one variable.Rules for solving linear equationsRule 1: Same quantity can be added to both sides of the equation.Rule 2: Same quantity can be subtracted from both sides of the equation.Rule 3: Both sides of an equation can be multiplied by the same number.Rule 4: Both sides of an equation can be divided by the same non zero number.Transposition MethodWhen any term or equation is shifted from one side to another side, then the sign of the term changes. This is known as transposition. The changes that takes place are:

Addition Changes to subtraction

Example : x+8 =11 x = 11-8 (+8 becomes 8 when shifted) x = 3Subtraction Changes to Addition.

Example: y-9 = 11 y= 11+9 (-9 becomes +9 when shifted) y=20

Multiplication Changes to division

Example: 5* = 20 5 * x = 20 (x is multiplied by 5 which on shifting 20 divides by 5) x= 20/5 x = 4Division changes to Multiplication

Example: x/6 = 2 x= 2 X 6 (Division by 6 changes to multiplication by 6 on shifting) x = 12 Examples1. Solve for x: 5x 2 = 3x 4

Solution:

5x -2 = 3x -45x = 3x 4 +25x 3x = -4 +22x = -2x = -2 / 2x = -1

2. Solve for x :2x x = 31 + 3 (2x+1)

Solution2x-x = 31 + 3 (2x +1)x= 31 + 6x +3x 6x = 31 + 3-5x = 34 x = - 34 / 5

Solve for x: 6(3x+2) -5 (6x-1) = 6(x-3)-5(7x-6) +12x

Solution 6(3x+2) -5 (6x-1) = 6(x-3)-5(7x-6) +12x18x +12 -30x +5 = 6x -18 -35x +30 +12x18x-30x-6x+35x-12x = -18+30-12-518x+35x -30x -12x-6x = 30-3553x-45x = -55x =-5x =-1Application of Linear Equation to word problemsFollow the steps given here to solve word problems successfully.

Read the problem thoroughly.Note what is given and what need to be find outDenote the unknown quantity with any literal, say x,y,z etc.Translate the statement of the given problem in to algebraic equation.Solve the equation for the unknownThe solution of an equation becomes the value of unknown.

ExamplesProblem : If three less then a number is 10 find the number

Solution : let the number be x so x-3 = 10 x = 10+3 x = 13Problem : A is twice as old as B. Three years ago As age was three times as of B find the age of A.

Solution : let Bs age be x years, So As age = 2x Three years ago Bs Age = x-3 As Age = 2x-3According to given condition2x-3 = 3(x-3)2x-3 = 3x-92x 3x = -9 + 3-x = -6x = 6

Bs age = 6 yearsAs age = 2*6 = 12 Yearsmixed word problemsProblem : One Number is 6 time the other. Their sum is 140 find the two numbers

Solutionlet the other number be xthen first number = 6xAccording to question 6x +x =1407x =140x = 140/7x =20

First number = 6 *20 = 120Other number = 20Thus 120 & 20 are required two numbersProblem : Gauri has a piggy bank it is full of 1 rupee and 50 paisa coins it contains three times as many 50 paisa coins as 1 rupee coins. The total amount in the piggy bank is 35 rupees how many coins are there of each kind in the piggy bank

Solution: let the number of 1 rupee be x then the number of 50 paisa coins = 3x rupees 35 = 35 *100 paisa = 3500 paisa Rs. 1 = 100 paisa and x coins make 100x paisa coins of 50 paisa are 3x X 50 = 150 x paisaTotal 250x paisa

According to Question 250x = 3500 x = 3500/ 250x= 14

Number of 1 Rupee Coin = 14Number of 50 paisa coins = 3x = 3 X 14 = 42.

Problem: The length of a rectangle is 6m less than three times its breadth. Find the length and the breadth of the rectangle if its perimeter is 148m.

Solution: Let the breadth of given rectangle be x m.Then , Length =(3x-6)mAccording to the question,Perimeter of Rectangle = 148m2(3x-6+x)=1482(4x-6)=1488x-12=1488x=148+128x=160x=160/2=80

Problem: A 100 litre solution of acid and water contains 20 litres of acid. How many water must be added to make the solution 16% acidic?

Solution: Let x litres of water be added to make the solution 16 % acidic.Then, the volume of solution = 100+x litres 16% of this is acidi.e,16/100(100+x) = 20 litres16(100+x)=20*1001600+16x=200016x=2000-1600=400 x=400/16=25 litres

Problem: A number consists of 2 digits whose sum is 8. if 18 is added to it its digits are reversed. Find the number.

Solution : Let the digit of the units place be x. then, the number at tens place=8-x original number = 10(8-x) +x = 80-10x+x =80-9xThe reversed number = 10x+8-x =9x+8According to question,80-9x+18 = 9x+8 9x+9x=80+18-8 18x=90 x=90/18=5Digit at units place = x = 5And, digit at tens place = 8-x =8-5 =3The number = 10*3+5 =30+5 =35 THANK

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