Problem 20 - Precalculus: Mathematics for Calculus

Question

Find the complete solution of the linear system, or show that it is inconsistent. \(\left\\{\begin{array}{rr}x-2 y+3 z= & -10 \\ 3 y+z= & 7 \\ x+y-z= & 7\end{array}\right.\).$

Step-by-Step Solution

Answer

The solution is \( x = 2, y = 3, z = -2 \).

Step 1: Write the system of equations

The given system of equations is: 1. \( x - 2y + 3z = -10 \) 2. \( 3y + z = 7 \) 3. \( x + y - z = 7 \).

Step 2: Express one variable in terms of others

From equation (2), express \( z \) in terms of \( y \): \( z = 7 - 3y \).

Step 3: Substitute the expression into other equations

Substitute \( z = 7 - 3y \) into equation (1) and equation (3). For equation (1): \( x - 2y + 3(7 - 3y) = -10 \) \( x - 2y + 21 - 9y = -10 \) \( x - 11y = -31 \) For equation (3): \( x + y - (7 - 3y) = 7 \) \( x + y - 7 + 3y = 7 \) \( x + 4y = 14 \).

Step 4: Solve the reduced system

We now have two equations with two variables: 1. \( x - 11y = -31 \) 2. \( x + 4y = 14 \). Subtract the first equation from the second: \((x + 4y) - (x - 11y) = 14 - (-31) \) \( 15y = 45 \) \( y = 3 \).

Step 5: Substitute back to find other variables

Now that \( y = 3 \), substitute back into the equations for \( x \) and \( z \).Using \( x - 11y = -31 \): \( x - 11(3) = -31 \) \( x - 33 = -31 \) \( x = 2 \).Using \( z = 7 - 3y \): \( z = 7 - 3(3) \) \( z = 7 - 9 \) \( z = -2 \).

Step 6: Verify the solution

Substitute \( x = 2, y = 3, z = -2 \) back into the original equations to verify they are satisfied:1. \( 2 - 2(3) + 3(-2) = 2 - 6 - 6 = -10 \) 2. \( 3(3) + (-2) = 9 - 2 = 7 \) 3. \( 2 + 3 - (-2) = 2 + 3 + 2 = 7 \).All equations are satisfied, confirming the solution.

Key Concepts

Solution of Linear Equations

When dealing with linear systems of equations, the primary goal is to find the values of unknown variables that satisfy all given equations. These equations are formed by expressions where each term is either a constant or the product of a constant and a single variable. In a linear system such as the one given:

• Equation 1: \( x - 2y + 3z = -10 \)
• Equation 2: \( 3y + z = 7 \)
• Equation 3: \( x + y - z = 7 \)

we need to determine the values of \( x \), \( y \), and \( z \) that work together to make all these equations true. Such a set of values is known as the solution to the system. If a solution exists such that it satisfies all equations simultaneously, we say the system is consistent. If no such solution exists, the system is called inconsistent.

Substitution Method

The substitution method is a powerful technique for solving systems of equations, especially when dealing with systems that involve more than two variables. This method involves solving one of the equations for one variable in terms of the others and then substituting this expression into the remaining equations. In our system:

• We started by expressing \( z \) in terms of \( y \) from the second equation: \( z = 7 - 3y \).
• This expression was then substituted into the other two equations, reducing the original system to a simpler one with only two variables.
• After substitution, the system simplifies to:
• \( x - 11y = -31 \)
• \( x + 4y = 14 \)

This approach reduces the number of variables step by step, making it easier to solve for the remaining variables. The method is particularly useful in breaking down complex systems into simpler, more manageable equations.

Verification of Solutions

In mathematics, verifying the solution of a linear system is a crucial final step to ensure the accuracy of the answers obtained through the substitution method or any other solving method. For our system:

• We found that \( x = 2 \), \( y = 3 \), and \( z = -2 \) satisfy all the original equations.
• Verification involves substituting these values back into the original equations to check if they hold true.
• For the first equation: \( 2 - 2(3) + 3(-2) = -10 \)
• For the second equation: \( 3(3) + (-2) = 7 \)
• For the third equation: \( 2 + 3 - (-2) = 7 \)

All equations satisfied confirm that the solution is indeed correct. This step is essential as it validates that no errors were made during calculation and ensures the solution's reliability.