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The sum of a two digit number and the number obtained by reversing the digits is 66. If the digits of the number differ by 2, find the number. How many such numbers are there?
There are two such numbers: 24 and 42.
Let’s denote the two-digit number as 10a + b, where a is the digit in the tens place and b is the digit in the ones place. The number obtained by reversing the digits is 10b + a.
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According to the given conditions:
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The sum of the original number and the number obtained by reversing the digits is 66, so we have the equation: (10a + b) + (10b + a) = 66
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The digits of the number differ by 2, so we have the equation: |a – b| = 2
Now, let’s solve this system of equations:
From the second equation, we have two cases:
- a – b = 2
- b – a = 2
Case 1: a – b = 2
Substitute a = b + 2 into the first equation: (10(b + 2) + b) + (10b + (b + 2)) = 66 (10b + 20 + b) + (10b + b + 2) = 66 22b + 22 = 66 22b = 44 b = 2 a = b + 2 = 4
So, one such number is 42.
Case 2: b – a = 2
Substitute b = a + 2 into the first equation: (10a + (a + 2)) + (10(a + 2) + a) = 66 (10a + a + 2) + (10a + 20 + a) = 66 22a + 22 = 66 22a = 44 a = 2 b = a + 2 = 4
So, another such number is 24.
Hence, there are two such numbers: 24 and 42.
Systems of Linear Equations in Algebra
A system of linear equations is a collection of two or more linear equations involving the same set of variables. The general form of a linear equation in two variables x and y is:
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ax + by = c
Where a, b, and c are constants, and a and b are not both zero. The solution to a linear equation is the set of values for the variables that make the equation true.
A system of linear equations involves multiple linear equations with the same variables. For example, a system of two linear equations in two variables x and y can be represented as:
ax + by = c dx + ey = f
A solution to a system of linear equations is a set of values for the variables that simultaneously satisfy all equations in the system. There are three possible scenarios for the solution of a system of linear equations:
- Consistent System: A system of equations that has at least one solution is called consistent.
- Inconsistent System: A system of equations that has no solution is called inconsistent.
- Dependent System: A system of equations that has infinitely many solutions is called dependent.
There are various methods to solve systems of linear equations, including:
- Graphical Method: Graphing each equation on the same coordinate plane and finding the point(s) of intersection, if any.
- Substitution Method: Solving one equation for one variable and substituting the expression into the other equation.
- Elimination Method: Adding or subtracting equations to eliminate one variable and then solving for the remaining variable.
- Matrix Method (or Gaussian Elimination): Representing the system of equations in matrix form and performing row operations to solve for the variables.
Each method has its advantages and is suitable for different situations. The choice of method often depends on the specific characteristics of the system of equations and personal preference.
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