Solve the following Word Problem on Linear Equation System

If three times the larger of the two numbers is divided by the smaller, then the quotient is 4 and remainder is 5. If six times the smaller is divided by the larger, the quotient is 4 and remainder is 2. Find the numbers.

Solution to Example 1 of Linear Equation System :

Let x be the larger and y be the smaller numbers. By data, if three times the larger of the two numbers is divided by the smaller, then the quotient is 4 and remainder is 5. i.e. if 3x is divided by y, the quotient is 4 and remainder is 5. ⇒ 3x = 4y + 5 ⇒ 3x - 4y = 5........................................(i) Also by data, If six times the smaller is divided by the larger, the quotient is 4 and remainder is 2. i.e. if 6y is divided by x, the quotient is 4 and remainder is 2. ⇒ 6y = 4x + 2 ⇒ -4x + 6y = 2........................................(ii)

Equations (i) and (ii) are the Linear Equations in two variables formed by converting the given word statements to the symbolic language.

Now we have to solve these simultaneous equations. To solve (i) and (ii), Let us make the coefficients of y the same. (i) x 3 gives 9x - 12y = 15........................................(iii) (ii) x 2 gives -8x + 12y = 4........................................(iv) (iii) + (iv) gives 9x - 8x = 15 + 4 ⇒ x = 19. Using this in (ii), we get -4(19) + 6y = 2 ⇒ -76 + 6y = 2 ⇒ 6y = 2 + 76 = 78 ⇒ y = 78⁄6 = 13. Thus the numbers are 19 and 13. Ans.

Check: three times the larger ( = 3 x 19 = 57) divided by smaller ( = 13) gives quotient 4 and remainder 5 ( since 4 x 13 = 52 and 57 - 52 = 5) (verified.) six times the smaller ( = 6 x 13 = 78) divided by larger ( = 19) gives quotient 4 and remainder 2 ( since 4 x 19 = 76 and 78 - 76 = 2) (verified.)

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Solve the following Word Problem Linear Equation System

The denominator of a fraction is greater than its numerator by 9. If 7 is subtracted from both its numerator and denominator, the fraction becomes 2⁄3. Find the original fraction.

Solution to Example 2 of Linear Equation System :

Let x be the numerator and y be the denominator. By data, the denominator is greater than its numerator by 9. ⇒ y = x + 9 ⇒ -x + y = 9.....................................(i) Also by data, If 7 is subtracted from both its numerator and denominator, the fraction becomes 2⁄3. ⇒ (x - 7)⁄(y - 7) = 2⁄3 Crossmultiplying, we get 3(x - 7) = 2(y - 7) ⇒ 3x - 21 = 2y - 14 ⇒ 3x - 2y = -14 + 21 ⇒ 3x - 2y = 7.....................................(ii)

Equations (i) and (ii) are the Linear Equations in two variables formed by converting the given word statements to the symbolic language.

Now we have to solve these simultaneous equations. To solve (i) and (ii), let us make the coefficients of y the same. (i) x 2 gives -2x + 2y = 18.....................................(iii) Adding (ii) and (iii), we get 3x - 2x = 7 + 18 ⇒ x = 25. Using this in (i), we get -25 + y = 9 ⇒ y = 9 + 25 = 34. The required fraction = 25⁄34. Ans.

Check: Since 34 = 25 + 9, "the denominator is greater than its numerator by 9" is satisfied.(verified.) Since (25 - 7)⁄(34 - 7) = 18⁄27 = 2⁄3, "If 7 is subtracted from both its numerator and denominator, the fraction becomes 2⁄3" is satisfied.(verified.)

Example 3 of Linear Equation System

Solve the following Word Problem on Linear Equation System

A and B each have a certain number of marbles. A says to B, " if you give 30 to me, I will have twice as many as left with you." B replies "if you give me 10, I will have thrice as many as left with you." How many marbles does each have?

Solution to Example 3 of Linear Equation System :

Let x be the number of marbles A has. And Let y be the number of marbles B has. If B gives 30 to A, then A has x + 30 and B has y - 30. By data, When this happens, A has twice as many as left with B. ⇒ x + 30 = 2(y - 30) = 2y - 2 x 30 = 2y - 60. ⇒ x - 2y = -60 - 30 ⇒ x - 2y = -90 ..........(i) If A gives 10 to B, then A has x - 10 and B has y + 10. By data, When this happens, B has thrice as many as left with A. ⇒ y + 10 = 3(x - 10) = 3x - 3 x 10 = 3x - 30 ⇒ y - 3x = -30 -10 ⇒ 3x - y = 40 ...........(ii)

Equations (i) and (ii) are the Linear Equations in two variables formed by converting the given word statements to the symbolic language.

Now we have to solve these simultaneous equations. To solve (i) and (ii), Let us make y coefficients same. (ii) x 2 gives 6x - 2y = 80 ...........(iii) x - 2y = -90 ..........(i) Subtracting 5x = 80 - (-90) = 80 + 90 = 170 ⇒ x = 170⁄5 = 34. Using this in Equation (ii), we get 3(34) - y = 40 ⇒ 102 - y = 40 ⇒ -y = 40 - 102 = -62 ⇒ y = 62. Thus A has 34 marbles and B has 62 marbles. Ans.

Check: If B gives 30 to A from his 62, then A has 34 + 30 = 64 and B has 62 - 30 = 32. Twice 32 is 64. (verified.) If A gives 10 to B from his 34, then A has 34 - 10 = 24 and B has 62 + 10 = 72. Thrice 24 is 72. (verified.)

Thus the Solution to Example 3 of Linear Equation System is verified.

Exercise on Linear Equation System

Solve the following Word Problems on Linear Equation System.

Five years hence, a man's age will be three times his son's age and five years ago, he was seven times as old as his son. Find their present ages.

If the length and breadth of a room are increased by 1 m each, its area is increased by 21 m^{2}. If the length is increased by 1 m and breadth decreased by 1m, the area is decreased by 5 m^{2}. Find the area of the room.

The students of a class are made to stand in complete rows. If one student is extra in each row, there would be 2 rows less, and if one student is less in each row, there would be 3 rows more. Find the number of students in the class.

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