| An equation of one variable and of first order (i.e., its highest power is one) is called a Linear equation. Such an equation has only one solution. A solution is also called the 'root' of the given equation. |
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Solve  |
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| Multiplying throughout by the LCM of the denominators (which is 12): |
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| 3x - 2 (x - 3) = 12 |
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| 3x - 2x + 6 = 12 |
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| x = 12 - 6 |
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 x = 6 |
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Solve:  |
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R.H.S  |
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| By cross multiplying: |
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| 3 (x2 - 5x + 6) = (x - 1) (3x - 8) |
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| 3x2 - 15x + 18 = 3x2 - 8x - 3x + 8 |
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| 3x2 - 15x + 18 = 3x2 - 11x + 8 |
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| 3x2 - 15x - 3x2 + 11x = 8 - 18 |
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| -4x = - 10 |
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| 4x = 10 |
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| Divide 40 into two parts so that one-third of the greater part may be equal to half of the lesser one. |
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| Let the greater part be x. |
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 the lesser part is 40 - x. |
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 of the greater =  of the lesser |
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 2x = 3 (40 - x) |
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 2x = 120 - 3x |
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 5x = 120 |
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 the two parts are 24 and 16. |
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| Mr.R is seven times as old as his son. 10 years hence, he will be three times as old as his son. Find their ages. |
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| Let the son's age be x years. |
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Mr.R's age is 7x years. |
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| After 10 years, his son's age will be = (x + 10) years |
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| After 10 years, Mr.R's age will be = (7x + 10) years |
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| By the given condition of the problem |
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| (7x + 10) = 3(x + 10) |
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 7x + 10 = 3x + 30 |
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 7x - 3x = 30 - 10 |
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 4x = 20 or x = 5 |
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 Son's age is 5 years. |
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Mr.R's age is 7  5 = 35 years |
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| Mr.R's age is 35 years and his son's age is 5 years. |
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Find a number, the half of which when added to 3, is equal to 12 diminished by  that number. |
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| Let x be the required number. |
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| The value of x is determined by solving the equation. |
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Thus   |
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| \ 2x + x = 36 |
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| \ 3x = 36 |
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| \ x = 12 |
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| \ The number is 12. |
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| A man walking at the rate of four kilometres an hour, covers a certain distance in three hours less than another who walks at the rate of three kilometres an hour. Find the distance. |
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As the distance is unknown, let x be the distance in kilometres and  is the time in hours taken by the 1st walker to cover the distance x.  is the time in hours taken by the 2nd walker to cover the same distance x. |
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 According to the terms of the problem,  |
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 4x - 3x = 36 |
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| or x = 36 km |
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| Distance is 36 km. |
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