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

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

To solve the problem of finding the two-digit number where the sum of its digits is 9, and nine times this number is twice the number obtained by reversing the order of the digits, follow these steps.

Let x be the ten's digit of the number.

Let y be the one's digit of the number.

Thus, the two-digit number can be written as 10x+y.

Given Conditions

The sum of the digits is 9: x+y=9 (Equation I)

Nine times the original number is equal to twice the number obtained by reversing the digits:

9(10x+y)=2(10y+x)

Simplify the Equation

Expand and simplify the second equation:

90x+9y=20y+2x

88x−11y=0(Equation II)

Solve the System of Equations

Solve Equation I and Equation II simultaneously. Start with Equation II:

88x=11y

8x=y(Equation III)

Substitute Equation III into Equation I:

x+8x=9

9x=9

x=1

Use the value of x to find y from Equation III:

y=8x

y=8(1)

y=8

The ten's digit is x=1 and the one's digit is y=8

Therefore, the desired number is 18.

System of Linear Equations

A system of linear equations involves two or more linear equations with multiple variables. The goal is to find values for these variables that satisfy all the given equations simultaneously. This concept is fundamental in algebra and forms the basis for understanding more complex mathematical relationships.

The problem of finding a two-digit number where the sum of its digits is 9, and nine times this number is twice the number obtained by reversing the order of the digits, is an example of solving a system of linear equations.

Steps in Solving Systems of Linear Equations

• Write down all the linear equations involved in the system.
• Decide whether to use the Graphical Method, Substitution Method, Elimination Method, or Matrix Method.

Graphical Method

• Plot each equation on a graph.
• Identify the point(s) where the lines intersect. This point(s) represents the solution to the system.

Substitution Method:

• Solve one of the equations for one variable in terms of the others.
• Substitute this expression into the other equation(s) to find the value of one variable.
• Substitute back to find the value of the other variable(s).

Elimination Method:

• Align the equations and multiply them if necessary to obtain coefficients that can be canceled.
• Add or subtract the equations to eliminate one variable.
• Solve the resulting equation for one variable.
• Substitute this value into one of the original equations to find the other variable(s).

Matrix Method:

• Write the system of equations in matrix form as AX=B.
• Use matrix operations to solve for X (the vector of variables), typically using methods such as Gaussian elimination or finding the inverse of matrix A.

Check Consistency:

Verify if the system is consistent (has at least one solution), inconsistent (has no solution), or dependent (has infinitely many solutions).

Interpret the Solution:

• Determine the solution in the context of the problem.
• Interpret the meaning of the solution in terms of the original variables.

Verify the Solution:

Substitute the solution back into the original equations to ensure that all equations are satisfied.