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adridgarcia
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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?
Prepare thyself for more questions.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 \( z = 60 \)
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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.
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.
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.
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.
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.