Problem 15 - A Graphical Approach to Precalculus with Limits

Question

Solve each system analytically. If the equations are dependent, write the solution set in terms of the variable \(z\). $$\begin{array}{r} x+3 y+4 z=14 \\ 2 x-3 y+2 z=10 \\ 3 x-y+z=9 \end{array}$$

Step-by-Step Solution

Answer

The solution is \((x, y, z) = (2, 0, 3)\).

Step 1: Write the system of equations

The given system of equations is: \[ \begin{align*} x + 3y + 4z &= 14 \ 2x - 3y + 2z &= 10 \ 3x - y + z &= 9 \end{align*} \] Our goal is to solve this system analytically.

Step 2: Eliminate y from Equations 1 and 2

First, let's consider Equation 1: \( x + 3y + 4z = 14 \) and Equation 2: \( 2x - 3y + 2z = 10 \). We can add these equations to eliminate \( y \): \[ \begin{align*} (x + 3y + 4z) + (2x - 3y + 2z) &= 14 + 10 \ 3x + 6z &= 24 \ x + 2z &= 8 \end{align*} \] We have one equation: \( x + 2z = 8 \).

Step 3: Eliminate y from Equations 1 and 3

Now, let's eliminate \( y \) from Equations 1 and 3. Multiply Equation 3 by 3 to align \( y \); \( 3x - y + z = 9 \) becomes \( 9x - 3y + 3z = 27 \). Add this result to Equation 1: \[ \begin{align*} (x + 3y + 4z) + (9x - 3y + 3z) &= 14 + 27 \ 10x + 7z &= 41 \end{align*} \] We have \( 10x + 7z = 41 \).

Step 4: Solve for x using simplified system

Now, use the two new equations: \( x + 2z = 8 \) and \( 10x + 7z = 41 \). Express \( x \) from \( x + 2z = 8 \): \( x = 8 - 2z \). Substitute this into \( 10x + 7z = 41 \): \[ \begin{align*} 10(8 - 2z) + 7z &= 41 \ 80 - 20z + 7z &= 41 \ -13z &= -39 \ z &= 3 \end{align*} \] With \( z = 3 \), substitute back: \( x = 8 - 2(3) = 2 \).

Step 5: Solve for y

Using \( x = 2 \) and \( z = 3 \), substitute into any original equation to find \( y \). Use Equation 1: \( x + 3y + 4z = 14 \): \[ \begin{align*} 2 + 3y + 4(3) &= 14 \ 2 + 3y + 12 &= 14 \ 3y &= 0 \ y &= 0 \end{align*} \] The solution is \( x = 2, y = 0, z = 3 \).

Step 6: Write the solution set

The solution to the system of equations is \( (x, y, z) = (2, 0, 3) \). Since the solution is a single point and not dependent on \( z \), there is no parameterization of \( z \).

Key Concepts

Dependent Equations

Dependent equations in a system are intriguing cases where one equation is a scalar multiple or a linear combination of another. This means there's a replica in disguise.
For example, imagine two equations that, when simplified, convey the same information.
If this is the case within a system of equations, it might suggest that there are infinite solutions rather than just one distinct solution.

• Dependent: One equation is a 'shadow' of another, potentially hinting at overlapping solutions.
• Not inherently solvable uniquely without additional constraints or data.

During the step-by-step process, the given equations are reduced without revealing such overlap, indicating distinct solutions in this context. Always keep an eye on whether your equations tell identical stories - this points to dependency.

Analytical Solution

An analytical solution involves finding a precise answer using algebraic and logical manipulations, ensuring each step builds upon the last.
This method encompasses using algebra, logic, and systematic reduction or elimination of variables. You start with equations manipulation to isolate variables.

• Exact answers: No estimation or graphical methods involved.
• Strategy: Simplify and solve equations using logical steps.

In the provided system, the solution is derived in a series of calculated maneuvers that progressively simplify the relationships among \(x\), \(y\), and \(z\), leading us to a concrete solution.

Elimination Method

The elimination method, a time-tested technique, is akin to solving a puzzle by gradually removing extraneous pieces. It involves adding or subtracting equations to cancel out one or more variables, enabling you to solve for the remaining variables.
This method was aptly applied in the step-by-step solution by:

• Combining equations to eliminate \(y\), yielding simpler equations in terms of \(x\) and \(z\).
• Further manipulating these reduced equations until all variables are isolated.

The art of elimination simplifies solving systems by transforming them into a more approachable state. Mastering elimination paves the way for tackling complex systems efficiently.

Solution Set

A solution set is a collection of all possible solutions that satisfy the given system of equations. In our context, it's what we derive once the system is solved.
For a linear system, this can either be:

• A unique solution, as seen here with \((2, 0, 3)\).
• No solution, if the equations are inconsistent.
• Infinite solutions when the equations are dependent and form a single line or plane.

Understanding the nature of the solution set is crucial. It not only reflects the problem-solving process but also the nature of the system's interrelations. Here, since the equations were consistent and non-dependent, the solution set is a singular point.