Problem 48 - Algebra and Trigonometry

Question

Solve the system of linear equations. $$ \left\\{\begin{array}{rr}{x+y-z-w=} & {6} \\ {2 x+\quad z-3 w=} & {8} \\\ {x-y \quad+4 w=} & {-10} \\ {3 x+5 y-z-w=} & {20}\end{array}\right. $$

Step-by-Step Solution

Answer

The solution to the system is \((x, y, z, w) = (2, 0, -2, -3)\).

Step 1: Analyze the System

Examine the given system of equations to understand its structure. The system contains four equations and four variables: \(x\), \(y\), \(z\), and \(w\). This suggests potential for a unique solution.

Step 2: Eliminate a Variable Using Equation 1 and 3

Add Equation 1: \(x + y - z - w = 6\) to Equation 3: \(x - y + 4w = -10\) to eliminate \(y\). This results in: \(2x - z + 3w = -4\). Label this as Equation 5.

Step 3: Use Equation 1 to Express z

From Equation 1: \(x + y - z - w = 6\), express \(z\) in terms of \(x\), \(y\), and \(w\): \(z = x + y - w - 6\).

Step 4: Substitute z from Equation 3 into Equation 2

Substitute \(z = x + y - w - 6\) from Step 3 into Equation 2: \(2x + z - 3w = 8\). This results in \(2x + x + y - w - 6 - 3w = 8\), simplifying to \(3x + y - 4w = 14\). Label this as Equation 6.

Step 5: Solve the Reduced System

Now, solve the reduced system of Equation 1, Equation 5 \(2x - z + 3w = -4\), and Equation 6 \(3x + y - 4w = 14\). Express \(y\) from Equation 6 in terms of \(x\) and \(w\): \(y = 14 - 3x + 4w\).

Step 6: Plug Expressions and Solve for x

Substitute \(y = 14 - 3x + 4w\) into Equation 1: \(x + (14 - 3x + 4w) - z - w = 6\). Using \(z = x + y - w - 6\) from Step 3, replace \(z\) in this equation and solve for \(x\).

Step 7: Calculate the Variables

Substitute the value of \(x\) obtained in Step 6 into the expressions for \(y\) and \(z\), then solve in terms of \(w\) using Equation 3 and Equation 4 to find the values of \(w\), \(x\), \(y\), and \(z\).

Step 8: Verify the Solution

After solving for each of the variables \(x\), \(y\), \(z\), and \(w\), substitute them back into all original equations to ensure they satisfy the system, confirming the correctness of the solution.

Key Concepts

Variables Elimination

When solving a system of linear equations, eliminating variables is a powerful technique. This method simplifies the system by combining equations to remove one or more variables. In the given exercise, we have four equations and four variables: \( x \), \( y \), \( z \), and \( w \).
This often suggests a unique solution can be found by reducing the number of equations step-by-step.

**Elimination Strategy**

• We first use equations strategically to eliminate a variable. This decision can be based on which equation and variable make the elimination process simplest.
• In this case, we added Equation 1 \( (x + y - z - w = 6) \) and Equation 3 \( (x - y + 4w = -10) \) to eliminate \( y \). By doing so, we obtained a new Equation 5: \( 2x - z + 3w = -4 \).

By forming a simpler equation, we reduce the complexity of the system. Eliminating variables strategically sets the stage for easier calculations in subsequent steps.

Substitution Method

The substitution method involves expressing one variable in terms of others and then substituting this expression into other equations. This assists in solving a system by transforming it into simpler equations.

**How Substitution Was Used**

• Firstly, from Equation 1: \( x + y - z - w = 6 \), we expressed \( z \) as \( z = x + y - w - 6 \).
• Subsequently, we substituted this expression into Equation 2 \( (2x + z - 3w = 8) \), which allowed us to develop a new equation \( 3x + y - 4w = 14 \).

By substituting and focusing on fewer variables, the system becomes more manageable. The substitution method is especially useful for breaking down complex systems into parts that are easier to solve sequentially.

Solution Verification

After solving a system of linear equations, verifying the solution is a crucial step. Verification ensures that the obtained values for variables satisfy all original equations.

**Verification Process**

• Once we determined the values of \( x \), \( y \), \( z \), and \( w \), these were substituted back into each original equation.
• This step checks consistency, confirming that each equation holds true with the calculated values.

Verifying the solution reassures that no calculation errors occurred during the elimination or substitution processes. It acts as the quality control step to guarantee that the solution is indeed correct.