2019 20 l inear e quations in o ne v ariable 33 did

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2019-20
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L INEAR E QUATIONS IN O NE V ARIABLE 33 Did you observe how we simplified the form of the given equation? Here, we had to multiply both sides of the equation by the LCM of the denominators of the terms in the expressions of the equation. RHS = 6 x – 2 + 7 2 = 4 7 3 6 6 2 2 2 x x - + = + The equation is x + 14 = 6 x + 3 2 or 14 = 6 x x + 3 2 or 14 = 5 x + 3 2 or 14 – 3 2 = 5 x (transposing 3 2 ) or 28 3 2 - = 5 x or 25 2 = 5 x or x = 25 1 5 5 5 2 5 2 5 2 × × = = × Therefore, required solution is x = 5 2 . Check : LHS = = 25 25 25 2(5 7) 2( 2) 4 2 2 2 - - = - - = + = 25 8 33 2 2 + = RHS = = 26 7 33 2 2 + = = LHS. (as required) EXERCISE 2.5 Solve the following linear equations. 1. 1 1 2 5 3 4 x x - = + 2. 3 5 21 2 4 6 n n n - + = 3. 8 17 5 7 3 6 2 x x x + - = - Note, in this example we brought the equation to a simpler form by opening brackets and combining like terms on both sides of the equation. 2019-20
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34 M ATHEMATICS 4. 5 3 3 5 x x - - = 5. 3 2 2 3 2 4 3 3 t t t - + - = - 6. 1 2 1 2 3 m m m - - - = - Simplify and solve the following linear equations. 7. 3( t – 3) = 5(2 t + 1) 8. 15( y – 4) –2( y – 9) + 5( y + 6) = 0 9. 3(5 z – 7) – 2(9 z – 11) = 4(8 z – 13) – 17 10. 0.25(4 f – 3) = 0.05(10 f – 9) 2.7 Equations Reducible to the Linear Form Example 18: Solve 1 3 2 3 8 x x + = + Solution: Observe that the equation is not a linear equation, since the expression on its LHS is not linear. But we can put it into the form of a linear equation. We multiply both sides of the equation by (2 x + 3), x x x + + × + 1 2 3 2 3 ( ) = 3 (2 3) 8 x × + Notice that (2 x + 3) gets cancelled on the LHS We have then, x + 1 = 3 (2 3) 8 x + We have now a linear equation which we know how to solve. Multiplying both sides by 8 8 ( x + 1) = 3 (2 x + 3) or 8 x + 8 = 6 x + 9 or 8 x = 6 x + 9 – 8 or 8 x = 6 x + 1 or 8 x – 6 x = 1 or 2 x = 1 or x = 1 2 The solution is x = 1 2 . Check : Numerator of LHS = 1 2 + 1 = 1 2 3 2 2 + = Denominator of LHS = 2 x + 3 = 1 2 2 × + 3 = 1 + 3 = 4 This step can be directly obtained by ‘cross-multiplication’ Note that 2 x + 3 0 (Why?) 2019-20
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L INEAR E QUATIONS IN O NE V ARIABLE 35 LHS = numerator ÷ denominator = 3 4 2 ÷ = 3 1 3 2 4 8 × = LHS = RHS. Example 19: Present ages of Anu and Raj are in the ratio 4:5. Eight years from now the ratio of their ages will be 5:6. Find their present ages. Solution: Let the present ages of Anu and Raj be 4 x years and 5 x years respectively. After eight years. Anu’s age = (4 x + 8) years; After eight years, Raj’s age = (5 x + 8) years. Therefore, the ratio of their ages after eight years = 4 8 5 8 x x + + This is given to be 5 : 6 Therefore, 4 8 5 8 x x + + = 5 6 Cross-multiplication gives 6 (4 x + 8) = 5 (5 x + 8) or 24 x + 48 = 25 x + 40 or 24 x + 48 – 40 = 25 x or 24 x + 8 = 25 x or 8 = 25 x – 24 x or 8 = x Therefore, Anu’s present age = 4 x = 4 × 8 = 32 years Raj’s present age = 5 x = 5 × 8 = 40 years EXERCISE 2.6 Solve the following equations. 1. 8 3 2 3 x x - = 2. 9 15 7 6 x x = - 3. 4 15 9 z z = + 4. 3 4 2 2 – 6 5 y y + - = 5. 7 4 4 2 3 y y + - = + 6. The ages of Hari and Harry are in the ratio 5:7. Four years from now the ratio of their ages will be 3:4. Find their present ages. 7. The denominator of a rational number is greater than its numerator by 8. If the numerator is increased by 17 and the denominator is decreased by 1, the number obtained is 3 2 . Find the rational number.
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