MHB Help? Algebra 2 Math - Solve X,Y,Z

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The discussion focuses on solving a word problem involving three numbers whose sum is 95, with specific relationships between them. The equations derived are \( x + y + z = 95 \), \( y = x + 5 \), and \( z = 3y \). Using substitution, the values of the numbers are calculated as \( x = 15 \), \( y = 20 \), and \( z = 60 \). The solution process includes combining like terms and isolating variables to find the final answers. The thread concludes with a light-hearted remark about the addictive nature of seeking help.
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The sum of three numbers is 95. The second number is 5 more than the first. The third number is 3 times the second. What are the numbers?
 
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Begin by turning the word problem into a system of equations:

Let \( x + y + z = 95 \), \( y = x + 5 \), and \( z = 3y \).

You can now use Elimination, Substitution, or Matrices to solve. I will use substitution by taking the 2nd and 3rd equations and getting the 1st equation in terms of \( y \).

\( y = x+ 5 \) subtract \( 5 \) from both sides.
\( x = y - 5 \)

Substitute \( x = y - 5 \) and \( z = 3y \) into \( x + y + z = 95 \):

\( ( y -5 ) + y + (3y) = 95 \) combine like terms
\( 5y - 5 = 95 \) add \( 5 \) to both sides
\( 5y = 100 \) divide by \( 5 \) on both sides
\( y = 20 \)

Substitute \( y = 20 \) into \( x = y - 5 \):
\( x = (20) - 5 \) simplify
\( x = 15 \)

Substitute \( y = 20 \) into \( z = 3y \):
\( z = 3(20) \) simplify
\( z = 60 \)

ANSWER: \( x = 15 \), \( y = 20 \), and \( z = 60 \)
 
Beer soaked comment follows.
SquareOne said:
Begin by turning the word problem into a system of equations:

Let \( x + y + z = 95 \), \( y = x + 5 \), and \( z = 3y \).

You can now use Elimination, Substitution, or Matrices to solve. I will use substitution by taking the 2nd and 3rd equations and getting the 1st equation in terms of \( y \).

\( y = x+ 5 \) subtract \( 5 \) from both sides.
\( x = y - 5 \)

Substitute \( x = y - 5 \) and \( z = 3y \) into \( x + y + z = 95 \):

\( ( y -5 ) + y + (3y) = 95 \) combine like terms
\( 5y - 5 = 95 \) add \( 5 \) to both sides
\( 5y = 100 \) divide by \( 5 \) on both sides
\( y = 20 \)

Substitute \( y = 20 \) into \( x = y - 5 \):
\( x = (20) - 5 \) simplify
\( x = 15 \)

Substitute \( y = 20 \) into \( z = 3y \):
\( z = 3(20) \) simplify
\( z = 60 \)

ANSWER: \( x = 15 \), \( y = 20 \), and \( z = 60 \)
Prepare thyself for more questions.
Spoon feeding can be very addictive.
 
jonah said:
Prepare thyself for more questions.
Spoon feeding can be very addictive.
Spoon feeding can be very "additive." (Dance)

-Dan
 

Thread 'Area of Overlapping Squares'

Here is a little puzzle from the book 100 Geometric Games by Pierre Berloquin. The side of a small square is one meter long and the side of a larger square one and a half meters long. One vertex of the large square is at the center of the small square. The side of the large square cuts two sides of the small square into one- third parts and two-thirds parts. What is the area where the squares overlap?
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