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Q2(iii):
Form the pair of linear equations in the following problems, and find their solutions (if they exist) by the elimination method : (iii) The sum of the digits of a two-digit number is 9. Also, nine times this number is twice the number obtained by reversing the order of the digits. Find the number.
Solution :
Given:
1. A two-digit number where the sum of its digits is $9$.
2. Nine times the original number is equal to twice the number obtained by reversing the digits.
To Find:
The original two-digit number.
Step 1: Defining the Variables
Let the digit at the tens place be $x$ and the digit at the units place be $y$.
Since it is a two-digit number, the value of the number can be expressed as:
Original Number $= 10x + y$
When the digits are reversed, the new tens digit becomes $y$ and the new units digit becomes $x$.
Reversed Number $= 10y + x$
Step 2: Formulating the Equations
According to the first condition, the sum of the digits is $9$:
$x + y = 9$ --- (Equation 1)
According to the second condition, nine times the original number is twice the reversed number:
$9(10x + y) = 2(10y + x)$
$90x + 9y = 20y + 2x$
$90x - 2x + 9y - 20y = 0$
$88x - 11y = 0$
Dividing the entire equation by $11$ to simplify:
$8x - y = 0$ --- (Equation 2)
Step 3: Solving by the Elimination Method
We have the system of equations:
(1) $x + y = 9$
(2) $8x - y = 0$
To eliminate $y$, we add Equation 1 and Equation 2:
$(x + y) + (8x - y) = 9 + 0$
$x + 8x + y - y = 9$
$9x = 9$
$x = \frac{9}{9}$
$x = 1$
Step 4: Finding the value of $y$
Substitute $x = 1$ into Equation 1:
$1 + y = 9$
$y = 9 - 1$
$y = 8$
Step 5: Determining the Number
The tens digit $x = 1$ and the units digit $y = 8$.
Original Number $= 10x + y = 10(1) + 8 = 18$.
Verification:
Sum of digits: $1 + 8 = 9$ (Satisfied).
Nine times the number: $9 \times 18 = 162$.
Twice the reversed number: $2 \times 81 = 162$.
Since $162 = 162$, the solution is correct.
Final Answer: The two-digit number is 18.
More Questions from Class 10 Mathematics Pair of linear equations in two variable EXERCISE 3.3
- Q1(i): Solve the following pair of linear equations by the elimination method and the substitution method : (i) x + y = 5 and 2x – 3y = 4
- Q1(ii): Solve the following pair of linear equations by the elimination method and the substitution method : (ii) 3x + 4y = 10 and 2x – 2y = 2
- Q1(iii): Solve the following pair of linear equations by the elimination method and the substitution method : (iii) 3x – 5y – 4 = 0 and 9x = 2y + 7
- Q1(iv): Solve the following pair of linear equations by the elimination method and the substitution method : (iv) $\frac{x}{2} + \frac{2y}{3} = -1$ and $x - \frac{y}{3} = 3$
- Q2(i): Form the pair of linear equations in the following problems, and find their solutions (if they exist) by the elimination method : (i) If we add 1 to the numerator and subtract 1 from the denominator, a fraction reduces to 1. It becomes $\frac{1}{2}$ if we only add 1 to the denominator. What is the fraction?
- Q2(ii): Form the pair of linear equations in the following problems, and find their solutions (if they exist) by the elimination method : (ii) Five years ago, Nuri was thrice as old as Sonu. Ten years later, Nuri will be twice as old as Sonu. How old are Nuri and Sonu?
- Q2(iv): Form the pair of linear equations in the following problems, and find their solutions (if they exist) by the elimination method : (iv) Meena went to a bank to withdraw ` 2000. She asked the cashier to give her ` 50 and ` 100 notes only. Meena got 25 notes in all. Find how many notes of ` 50 and ` 100 she received.
- Q2(v): Form the pair of linear equations in the following problems, and find their solutions (if they exist) by the elimination method : (v) A lending library has a fixed charge for the first three days and an additional charge for each day thereafter. Saritha paid ` 27 for a book kept for seven days, while Susy paid ` 21 for the book she kept for five days. Find the fixed charge and the charge for each extra day.
CBSE Solutions for Class 10 Mathematics Pair of linear equations in two variable
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