Solve the following using system of equations. The sum of three numbers is 26. The middle number numerically is one more than three times the smallest number. The largest number is two times less than five times the smallest number. Find the three numbers.
x+y+z = 26
y = 3x+1
z = 5x-2
hat do you get?
Honestly, in English, the term xxx times less than is poor usage. It apparently comes from translation of a Polish language concept, and in the translation, it becomes confusing.
If you ask 10 math experts what "xxxx times less than" means, you will get about eight different answers.
It is hard for me to believe you got this assigned in a college course. Ask the instructor to translate it, then after the course, forget you learned it, no one in English speaking will ever use the term.
There are a number of internet conversations on this, and they (unless they originally spoke Polish) are not in uniformity on agreement.
To solve this problem using a system of equations, let's denote the three numbers as x, y, and z.
1) The sum of three numbers is 26:
x + y + z = 26
2) The middle number numerically is one more than three times the smallest number:
y = 3x + 1
3) The largest number is two times less than five times the smallest number:
z = 5x - 2
Now we have a system of three equations with three variables. We can solve this system by substitution or elimination.
Let's start by solving it using the substitution method:
1) From equation (2), we have y = 3x + 1.
2) Substitute y in equation (1) with 3x + 1:
x + (3x + 1) + z = 26
4x + 1 + z = 26 => 4x + z = 25, (equation 4)
3) Substitute y in equation (3) with 3x + 1:
z = 5x - 2
4) Substitute z in equation (4) with 5x - 2:
4x + (5x - 2) = 25
9x - 2 = 25
9x = 27
x = 3
Now that we have the value of x, we can substitute it back into the original equations to find y and z.
Substituting x = 3 into equation (2):
y = 3(3) + 1
y = 9 + 1
y = 10
Substituting x = 3 into equation (3):
z = 5(3) - 2
z = 15 - 2
z = 13
Therefore, the three numbers are x = 3, y = 10, and z = 13.