Problem 56 - Precalculus: Mathematics for Calculus

Question

Solving a Linear System Solve the system of linear equations. $$\left\\{\begin{array}{l} 3 x-y+2 z=-1 \\ 4 x-2 y+z=-7 \\ -x+3 y-2 z=-1 \end{array}\right.$$

Step-by-Step Solution

Answer

\( x = -2, y = 1, z = 3 \)

Step 1: Analyze the system

We have a system of three linear equations with three unknowns: \( x, y, \) and \( z \). The equations are as follows: 1. \( 3x - y + 2z = -1 \) 2. \( 4x - 2y + z = -7 \) 3. \( -x + 3y - 2z = -1 \)

Step 2: Eliminate one variable

First, we aim to eliminate one of the variables. Let's eliminate \( z \) from the first two equations. Multiply the second equation by 2 to align \( z \) terms: 1. \( 3x - y + 2z = -1 \) 2. \( 8x - 4y + 2z = -14 \) Subtract the first equation from the modified second equation to eliminate \( z \): \( (8x - 4y + 2z) - (3x - y + 2z) = -14 - (-1) \) This simplifies to: \( 5x - 3y = -13 \)

Step 3: Eliminate the same variable from a different pair

Now eliminate \( z \) from the second and third equations. Consider the equation from Step 2 as a separate system: Multiplying the third equation by 1 for alignment: \( -x + 3y - 2z = -1 \) Adding it to \( 8x - 4y + 2z = -14 \) to eliminate \( z \): \( (8x - 4y + 2z) + (-x + 3y - 2z) = -14 - 1 \) This simplifies to: \( 7x - y = -15 \)

Step 4: Solve the two-variable system

We now have two equations with two variables: \( 5x - 3y = -13 \) \( 7x - y = -15 \) Multiply the second equation by 3 to set up elimination: \( 21x - 3y = -45 \) Now, subtract the first new equation: \( (21x - 3y) - (5x - 3y) = -45 - (-13) \) This simplifies to: \( 16x = -32 \) Solving for \( x \), we get \( x = -2 \).

Step 5: Substitute to find another variable

Now, substitute \( x = -2 \) back into one of the two-variable equations, say \( 7x - y = -15 \): \( 7(-2) - y = -15 \) This simplifies to: \( -14 - y = -15 \) So, \( y = 1 \).

Step 6: Solve for the remaining variable

Use the values of \( x \) and \( y \) to solve for \( z \) using any of the original equations. Let's use \( 3x - y + 2z = -1 \): \( 3(-2) - 1 + 2z = -1 \) Which simplifies to: \( -6 - 1 + 2z = -1 \) \( 2z = 6 \) So, \( z = 3 \).

Step 7: Verify the solution

Now, substitute \( x = -2 \), \( y = 1 \), and \( z = 3 \) back into all three original equations to ensure they are satisfied: 1. \( 3(-2) - 1 + 2(3) = -6 - 1 + 6 = -1 \) (True) 2. \( 4(-2) - 2(1) + 3 = -8 - 2 + 3 = -7 \) (True) 3. \( -(-2) + 3(1) - 2(3) = 2 + 3 - 6 = -1 \) (True) Since all equations work, the solution is verified.

Key Concepts

System of Equations

A system of equations consists of multiple equations with multiple unknown variables. In our problem, we have a system of three linear equations with three variables: \(x\), \(y\), and \(z\). Each equation represents a plane in a three-dimensional space, and the solution is the point where these planes intersect.
Understanding that we have three variables to find requires having at least three independent equations. Each equation provides a necessary relationship among the variables.
Here are the given equations from the problem:

• Equation 1: \( 3x - y + 2z = -1 \)
• Equation 2: \( 4x - 2y + z = -7 \)
• Equation 3: \( -x + 3y - 2z = -1 \)

Solving such a system involves finding the values of \(x\), \(y\), and \(z\) that satisfy all equations simultaneously. This is fundamental to solving linear systems, as each intersection point correlates to a possible solution set.

Variables Elimination

The elimination method is a strategy to solve a system of equations. It involves manipulating the equations to eliminate one of the variables, thus reducing the system's complexity.
In our exercise, the goal was to eliminate \(z\) from the equations. By aligning the equations to have the same coefficient for \(z\), then subtracting one equation from the other, we successfully remove that variable.
For example, the operation:

• Original Equation: \(8x - 4y + 2z = -14 \)
• Modified Equation: \(3x - y + 2z = -1 \)

Subtracting these gives us \(5x - 3y = -13\), a new equation without \(z\). This significantly simplifies the problem, making it easier to handle.
Repeating this process correctly will reduce the system step by step until you are left with two equations with two unknowns, and finally one equation with one unknown. Solving for that unknown will allow us to back substitute into the previous equations to find the remaining variables.

Verification of Solution

Verification is a crucial step when solving systems of equations. It ensures accuracy and confirms that the solution satisfies all original equations.
In our scenario, we verified by substituting the calculated values \( x = -2 \), \( y = 1 \), and \( z = 3 \) back into the original equations:

• First equation: \(3(-2) - 1 + 2(3) = -1\)
• Second equation: \(4(-2) - 2(1) + 3 = -7\)
• Third equation: \(-(-2) + 3(1) - 2(3) = -1\)

Each substitution yields a true statement, verifying our solution. It's important to always perform this step to ensure the solutions are correct, especially in more complex systems where errors can easily occur. It acts as a final check, reinforcing confidence in the results.