the sum of three numbers is 14 Three times the smallest is 1

the sum of three numbers is 14. Three times the smallest is 1 less than the largest, while the sum of the largest and smallest is 9. Use a linear system in three variables to find the three numbers. The three numbers are , , and .

Solution

Let x , y z be the three numbers and let x be the smallest and z be the biggest.

Then x + y + z = 14...(1) 3x = z - 1. or, z - 3x = 1..(2) and z + x = 9...(3)

On subtracting the 2nd equation from the 3rd equation, we get z + x - z + 3x = 9 - 1 or, 4x = 8 so that x = 2.

Now, from the 3rd equation, we have z + 2 = 9 so that z = 9 - 2 = 7. Also, from the 1st equation, we get y = 14 - x - z = 14 - 2 - 7 = 5. We can verify the result by using these values of x, y, z in the given statements. The sum of x,y, z is 14, 3*2 = 7 - 1 and 7 + 2 = 9.