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

In summary, the sum of three numbers is 95, with the second number being 5 more than the first, and the third number being 3 times the second. The numbers are x=15, y=20, and z=60.
  • #1
adridgarcia
1
0
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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  • #2
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 \( z = 60 \)
 
  • #3
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.
 
  • #4
jonah said:
Prepare thyself for more questions.
Spoon feeding can be very addictive.
Spoon feeding can be very "additive." (Dance)

-Dan
 

FAQ: Help? Algebra 2 Math - Solve X,Y,Z

What is the best approach to solving a system of equations with three variables?

The best approach to solving a system of equations with three variables (X, Y, Z) is to use the substitution method or the elimination method. Both methods involve isolating one variable and substituting it into the other equations to solve for the remaining variables.

How do I know if a system of equations has a unique solution?

A system of equations with three variables (X, Y, Z) has a unique solution if the number of equations equals the number of variables and the determinant of the coefficient matrix is non-zero. This means that the system has a distinct solution for each variable.

Can I solve a system of equations with three variables without using matrices?

Yes, it is possible to solve a system of equations with three variables without using matrices. The substitution and elimination methods can be used to solve the equations algebraically without the need for matrices.

How do I check my solution for a system of equations with three variables?

To check your solution for a system of equations with three variables (X, Y, Z), you can substitute the values of X, Y, and Z into each equation and see if they satisfy the equation. If all three equations are satisfied, then your solution is correct.

Can I use a graphing calculator to solve a system of equations with three variables?

Yes, you can use a graphing calculator to solve a system of equations with three variables (X, Y, Z). Most graphing calculators have a function to solve systems of equations, which can be accessed by entering the equations and variables into the calculator.

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