Problem 5 - Precalculus

Question

Solve each system of linear equations. $$\begin{array}{rr} -x+y-z= & -1 \\ x-y-z= & 3 \\ x+y-z= & 9 \end{array}$$

Step-by-Step Solution

Answer

The solution is \(x = 5\), \(y = 3\), \(z = -1\).

Step 1: Write the System of Equations

We start by clearly writing the system of equations: 1. \, \(-x + y - z = -1\)2. \, \(x - y - z = 3\)3. \, \(x + y - z = 9\)

Step 2: Label the Equations

For convenience, let's label each equation:- Equation (1): \(-x + y - z = -1\)- Equation (2): \(x - y - z = 3\)- Equation (3): \(x + y - z = 9\)

Step 3: Eliminate One Variable

To eliminate one variable, add Equation (1) and Equation (2):\((-x + y - z) + (x - y - z) = -1 + 3\)This simplifies to \(-2z = 2\), so \(z = -1\).

Step 4: Substitute \(z = -1\) into Equations

Substitute \(z = -1\) back into Equation (2):\(x - y - (-1) = 3\)This simplifies to \(x - y + 1 = 3\)Therefore, \(x - y = 2\) (Equation 4).

Step 5: Substitute \(z = -1\) in Another Equation

Substitute \(z = -1\) into Equation (3):\(x + y - (-1) = 9\)This simplifies to \(x + y + 1 = 9\)Therefore, \(x + y = 8\) (Equation 5).

Step 6: Solve the New System

Now, solve the system from Equations 4 and 5:- \(x - y = 2\)- \(x + y = 8\)Add them:\((x - y) + (x + y) = 2 + 8\)This becomes \(2x = 10\), so \(x = 5\).

Step 7: Find the Remaining Variable \(y\)

Substitute \(x = 5\) into Equation 5:\(5 + y = 8\)This results in \(y = 3\).

Step 8: State the Solution

The solution to the system of equations is \(x = 5\), \(y = 3\), and \(z = -1\).

Key Concepts

Elimination Method

The elimination method is a way to simplify a system of linear equations by removing one of the variables. This is achieved by adding or subtracting equations to cancel out a particular variable.
This technique is particularly useful when the coefficients of one of the variables in two different equations are opposites. In this exercise, we focused on eliminating the variable \(z\).
Here's how it works:

• Start with two equations that have the same variable, but with opposite signs.
• Add the two equations. This cancels out the variable you want to eliminate.
• Solve the resulting simpler equation. For instance, simplifying our equations gave \(-2z = 2\), resulting in \(z = -1\).

By eliminating variables step-by-step, we transformed our system into something that's easier to solve.

Substitution Method

The substitution method involves solving one equation for one of the variables and then plugging that expression into the other equations.
This method gives you a straightforward path to isolate and solve for other variables.
In this exercise:

• Once we found \(z = -1\), we substituted \(z\) back into the remaining equations. This helps to simplify and solve for another variable.
• Substituting \(z = -1\) into equations was straightforward because, once \(z\) was removed, equations became simpler. For example, from \(x + y + 1 = 9\) we found \(x + y = 8\).

Substitution is invaluable when solving a system as it reduces complexity by focusing on one variable at a time.

Solving Equations Step-by-Step

Breaking down equation solving into manageable steps is crucial in getting to the correct solution.
This approach ensures that each stage is understood before proceeding to the next, minimizing mistakes.
Here’s how the process unfolds:

• Write out the full system clearly. This helps keep track of each part of the system.
• Systematically eliminate variables using methods like elimination or substitution.
• Solve for one variable at a time, ensuring to substitute found values back into the equations.
• Continue simplifying and solving until all variables are found. Check your solution to ensure it's correct across all original equations.

Using the step-by-step method provides a structured path to solve even complex systems with ease.