The sum of the digits in a four-digit number is 10 . Twice the sum of the thousands digit and th

The sum of the digits in a four-digit number is 10 . Twice...

Key Concepts

Place Value

Place value refers to the value of a digit based on its position within a number. In a multi-digit number, such as a four?digit number, each digit occupies a specific place (thousands, hundreds, tens, and ones) that determines its contribution to the overall value of the number. This concept is crucial when setting up equations that involve individual digits of a number.

Digit Sum

The digit sum is the total obtained by adding all individual digits of a number. In problems that specify constraints related to the sum of digits, this concept is used to create an equation that represents this condition, providing a starting point for solving the problem.

System of Linear Equations

A system of linear equations is a collection of equations that can be solved simultaneously to find common values for their variables. In digit problems, each condition provided about the digits is translated into an equation, and the system is solved to determine the individual digits. This technique is essential in systematically handling multiple constraints.

Algebraic Representation and Substitution

Algebraic representation involves expressing the relationships between different digits using variables and equations. Substitution is then used to reduce the number of variables by replacing one variable with an expression involving another, based on given relationships. This method simplifies the process of solving the system of equations derived from the problem conditions.

Digit Constraints

Digit constraints refer to the inherent limitations on the values that digits can take (typically 0 through 9, with additional conditions for digits in specific positions, such as the thousands digit being nonzero in a four-digit number). These constraints ensure that the solutions obtained from the equations are valid as digits and help in narrowing down the possible values during the problem-solving process.

Transcript

00:01 For this question we will assume that the 4 digit number is wx y z where these are the these four variables represent the 4 digits in the number.

00:16 Now this question some might argue will be easier to solve with other methods of solving system of equations but since we are using the matrix method here we will solve the entire problem using a matrix method.

00:31 So the first equation we can write here is w plus x plus y plus z because they have said the sum of the digits is 10 so this is equal to 10.

00:42 Now from the next relation given in the question that is two times the sum of the thousands and tens digit number is equal to the sum of the hundreds and one's place digit minus one.

01:00 So if we simplify that relation, we get the equation 2w minus x plus 2y minus z is equal to minus 1.

01:11 Now the third relation given is that the 10th place digit is twice the 1th place digit.

01:20 So we'll write 2w minus y is equal to 0.

01:25 Similarly from the fourth relation given that the one's place digit is the sum of the digits in the thousands and the hundreds place we can write on rearranging w plus x minus z is equal to zero now we will write these four equations in matrix form so taking the coefficients and the constant terms we get the matrix one one one ten then then 2 minus 1, 2 minus 1, 2 minus 1, 2.

02:02 Now the coefficient for x is 0 and then minus 1 and 0, here is 0.

02:09 Finally 1 1, 0 and minus 1 and here we get 0.

02:14 So we draw dotted line to separate the coefficients from the constant terms.

02:19 Next we have to do a few row operations to turn this matrix into its row echelon form.

02:25 The first set of row operations we will do is row 2 prime or new row 2 is twice row 1 minus row 2.

02:35 We'll also do a operation on row 3.

02:38 So row 3 prime is twice r1 minus r3.

02:42 Finally we'll do an operation on row 4.

02:45 So row 4 prime is r1 minus r4...