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Defination<br />A Linear equation is an equation involving single variables or literals which have highest power 1, i.e. we have linear equation as one variable.<br />
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Rules for solving linear equations<br /><ul><li>Rule 1: Same quantity can be added to both sides of the equation.
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Rule 2: Same quantity can be subtracted from both sides of the equation.
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Rule 3: Both sides of an equation can be multiplied by the same number.
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Rule 4: Both sides of an equation can be divided by the same non zero number.</li></li></ul><li>Transposition Method<br />When 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:<br />Addition Changes to subtractionExample : x+8 =11 x = 11-8 (+8 becomes – 8 when shifted) x = 3<br />Subtraction Changes to Addition.Example: y-9 = 11 y= 11+9 (-9 becomes +9 when shifted) y=20<br />
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Multiplication Changes to divisionExample: 5* = 20 5 * x = 20 (x is multiplied by 5 which on shifting 20 divides by 5)<br /> x= 20/5 x = 4<br />Division changes to MultiplicationExample: x/6 = 2 x= 2 X 6 (Division by 6 changes to multiplication by 6 on shifting) x = 12 <br />
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Application of Linear Equation to word problems<br />Follow the steps given here to solve word problems successfully.<br /><ul><li>Read the problem thoroughly.
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Note what is given and what need to be find out
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Denote the unknown quantity with any literal, say x,y,z etc.
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Translate the statement of the given problem in to algebraic equation.
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The solution of an equation becomes the value of unknown.</li></li></ul><li>Examples<br />Problem : <br /> If three less then a number is 10 find the number<br />Solution : <br /> let the number be x <br /> so x-3 = 10<br /> x = 10+3<br /> x = 13<br />
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Problem : <br /> A is twice as old as B. Three years ago A’s age was three times as of B find the age of A.<br />Solution : <br />let B’s age be x years, So A’s age = 2x<br /> Three years ago B’s Age = x-3<br /> A’s Age = 2x-3<br />According to given condition<br /> 2x-3 = 3(x-3)<br /> 2x-3 = 3x-9<br /> 2x – 3x = -9 + 3<br /> -x = -6<br /> x = 6<br />B’s age = 6 years<br />A’s age = 2*6 = 12 Years<br />
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mixed word problems<br />Problem :<br /> One Number is 6 time the other. Their sum is 140 find the two numbers<br />Solution<br /> let the other number be x<br /> then first number = 6x<br />According to question <br /> 6x +x =140<br /> 7x =140<br /> x = 140/7<br /> x =20<br />First number = 6 *20 = 120<br />Other number = 20<br />Thus 120 & 20 are required two numbers<br />
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Problem : <br />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<br />Solution:<br /> let the number of 1 rupee be x<br /> then the number of 50 paisa coins = 3x<br /> rupees 35 = 35 *100 paisa = 3500 paisa<br /> Rs. 1 = 100 paisa and x coins make 100x paisa<br /> coins of 50 paisa are 3x X 50 = 150 x paisa<br />Total 250x paisa<br />According to Question 250x = 3500<br /> x = 3500/ 250<br /> x= 14<br />Number of 1 Rupee Coin = 14<br />Number of 50 paisa coins = 3x = 3 X 14 = 42.<br />
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Problem: <br /> 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.<br />Solution:<br /> Let the breadth of given rectangle be x m . <br />Then , Length =(3x-6)m<br />According to the question,<br />Perimeter of Rectangle = 148m<br />2(3x-6+x)=148<br />2(4x-6)=148<br />8x-12=148<br />8x=148+12<br />8x=160<br />x=160/2=80<br />
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Problem: <br /> 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?<br />Solution:<br /> Let x litres of water be added to make the solution 16 % acidic.<br />Then, the volume of solution = 100+x litres<br />16% of this is acid<br />i.e, 16/100(100+x) = 20 litres<br /> 16(100+x)=20*100<br /> 1600+16x=2000<br /> 16x=2000-1600=400<br /> x=400/16=25 litres<br />
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Problem:<br /> A number consists of 2 digits whose sum is 8. if 18 is added to it its digits are reversed. Find the number.<br />Solution :<br /> Let the digit of the units place be x.<br /> then, the number at tens place=8-x<br /> original number = 10(8-x) +x<br /> = 80-10x+x<br /> =80-9x<br />The reversed number = 10x+8-x<br /> =9x+8<br />According to question,<br /> 80-9x+18 = 9x+8<br /> 9x+9x=80+18-8<br /> 18x=90<br /> x=90/18=5<br /> Digit at units place = x = 5<br />And, digit at tens place = 8-x =8-5 =3<br /> The number = 10*3+5 =30+5<br /> =35 <br />
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