The sum of two numbers is 15 and the sum of their squares is 377. What is the larger number?
4 Answers
The larger number is
Explanation:
Write two equations with two variables:
Use substitution to solve:
-
Solve for one variable
# x = 15 - y# -
Substitute
# x = 15 - y# into the second equation:
#(15 - y)^2 + y^2 = 377# -
Distribute:
# (15-y)(15-y) + y^2 = 377#
#15^2 - 30 y +y^2 + y^2 = 377#
#255 - 30 y + 2y^2 = 377# -
Put in general form
#Ax^2 + Bx +C = 0# :
#2y^2 - 30y +225 - 377 = 0#
#2y^2 - 30y - 152 = 0# -
Factor
#2(y^2 - 15y - 76) = 0#
#2(y +4)(y - 19) = 0#
#y = -4, y = 19# -
Check:
#-4 + 19 = 15#
#(-4)^2 + 19^2 = 377#
The larger number is 19.
Explanation:
Since you have two numbers, you must have two equations that relate these numbers to one another. Each sentence provides one equation, if we can translate them properly:
"The sum of two numbers is 15" :
"The sum of their squares is 377" :
Now, we must use the simpler equation to replace one of the unknowns in the more complex equation:
Now, the second equation becomes
Expand the binomial:
Write in standard from:
This can be factored (because the determinant
Might be simpler to just use the quadratic formula, though:
If you check the two answers in the original equations, you will find that both yield that same result! The two numbers we seek are 19 and -4.
That is, if you put
If you put
This happens because it does not matter which value we use in the substitution. Both yield the same result.
Explanation:
let say the two numbers are
therefore the largest number is
Explanation:
It is possible to define both numbers by using only one variable.
The sum of two numbers is
If one number is
The sum of their squares is
Find factors of
The two numbers are: