Problem 1 :
The sum of two numbers is 7. The difference between 5 times the larger and 3 times the smaller is equal 11. Find the numbers.
Let 'x' and 'y' be the two numbers such that x > y.
Given : The sum of two numbers is 7.
x + y = 7 -----(1)
Given : The difference between 5 times the larger and 3 times the smaller is equal 11.
5x - 3y = 11 -----(2)
Solve (1) and (2) using elimination :
Eliminate one of the variables to get the value of the other variable.
In (1) and (2), both the variables 'x' and 'y' do not have the same coefficient.
One of the variables must have the same coefficient.
So, multiply both sides of (1) by 3 to make the coefficients of 'y' same.
(1) ⋅ 3 -----> 3x + 3y = 21 -----(3)
In (2) and (3), 'y' has the same coefficient with different signs.
Add (2) and (3) to eliminate the variable 'y' and solve for 'x'.
(2) + (3) :
8x = 32
Divide each side by 8.
x = 4
Substitute 4 for x in (1).
(1)-----> 4 + y = 7
Subtract 4 from each side.
y = 3
Therefore, the two numbers are 3 and 4.
Problem 2 :
The monthly incomes of A and B are in the ratio 3:4 and their monthly expenditures are in the ratio 5:7. If each saves ₹ 5,000 per month, find the monthly income of each.
Given : The monthly incomes of A and B are in the ratio 3 : 4.
Then, the monthly incomes of A and B can be assumed as 3x and 4x respectively.
Given : The monthly expenditures of A and B are in the ratio 5 : 7.
Then, the monthly expenditures of A and B can be assumed as 5y and 7y respectively.
The relationship between income, expenditure and savings is
Income - Expenditure = Savings
3x - 5y = 5000 ------(1)
4x - 7y = 5000 ------(2)
Because we have to find the monthly incomes of A and B, we need the value of x.
So, we have to eliminate 'y' in the above two equations.
In (1) and (2), 'y' does not have the same coefficient.
So, multiply both sides of (1) by 7 and (2) by -5 to make the coefficients of 'y' same with different signs in the equations.
(1) ⋅ 7 :
21x - 35y = 35000 ------(3)
(1) ⋅ -5 :
-20x + 35y = -25000 ------(4)
Add (3) and (4) to eliminate 'y' and find the value of 'x'.
x = 10000
Monthly income of A = 3x = 3(10000) = 30000
Monthly income of B = 4x = 4(10000) = 40000
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