Solving a System of Equations: Uncovering Two Special Numbers

Can you solve a mathematical puzzle where the difference of two numbers is 16 and the greater number is 5 less than 4 times the smaller number? By delving into the details of solving a system of equations, we can discover the solution.

Defining the Numbers

Let us define the two numbers as follows:

x is the smaller number. y is the greater number.

Based on the problem, we have two equations:

y - x 16 (The difference between the two numbers is 16) y 4x - 5 (The greater number is 5 less than 4 times the smaller number)

Solving the System of Equations

We can substitute the second equation into the first:

y - x 16

Substitute y with 4x - 5:

(4x - 5) - x 16

Simplify the equation:

4x - 5 - x 16

3x - 5 16

Now, add 5 to both sides:

3x 21

Divide by 3:

x 7

Now, use the second equation to find y:

y 4x - 5

Substitute x with 7:

y 4(7) - 5

y 28 - 5

y 23

Final Answer

The two numbers are:

Smaller number x 7 Greater number y 23

A Second Approach

Let's verify this solution with a different set of values: u and v also satisfy the conditions of the problem.

u - v 16 3u 7v

Again, by solving these equations:

u 28, v 12

Both sets of solutions (7, 23) and (28, 12) satisfy the conditions:

28 - 16 12 (or 12 16 28) 3 × 28 and 7 × 12 each multiply to 84

In conclusion, the methods and logic used in solving these equations are consistent and valid. The numbers that satisfy the given conditions are 23 and 28, as well as the alternative 7 and 12.

Keywords: system of equations, mathematical problem solving, algebraic equations.