Problem 46 - 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 \(z\). $$\begin{array}{r} x+3 y+z=6 \\ 3 x+y-z=6 \\ x-y-z=0 \end{array}$$

Step-by-Step Solution

Answer

The solution set is \( (\frac{3+z}{2}, \frac{3-z}{2}, z) \) for all real \( z \).

Step 1: Identify the System of Equations

We have the system of equations:1. \( x + 3y + z = 6 \)2. \( 3x + y - z = 6 \)3. \( x - y - z = 0 \).

Step 2: Solve the Third Equation for x

From the third equation \( x - y - z = 0 \), solve for \( x \): \[ x = y + z \].

Step 3: Substitute x in the First Equation

Substitute \( x = y + z \) into the first equation:\( (y + z) + 3y + z = 6 \).Simplify to get:\[ 4y + 2z = 6 \].

Step 4: Simplify Equation

Divide the equation by 2:\[ 2y + z = 3 \]. This is Equation (4).

Step 5: Substitute x in the Second Equation

Substitute \( x = y + z \) into the second equation:\( 3(y+z) + y - z = 6 \).Simplify to get:\[ 4y + 2z = 6 \].

Step 6: Analyze the System Equivalence

Both the derived equations from the substitution steps are essentially the same:\[ 2y + z = 3 \].This suggests that the original equations are dependent.

Step 7: Express the Solution Set in Terms of z

From Equation (4), express \( y \) in terms of \( z \):\[ y = \frac{3 - z}{2} \].Use \( x = y + z \) to express \( x \) in terms of \( z \):\[ x = \frac{3 - z}{2} + z = \frac{3 + z}{2} \].

Step 8: Solution Set Representation

The solution set of the system in terms of \( z \) is:\[ \left( \frac{3+z}{2}, \frac{3-z}{2}, z \right) \] for all real \( z \).

Key Concepts

Dependent Equations

To understand dependent equations, imagine equations that describe lines on a graph. If two equations are dependent, they essentially describe the same line, just expressed differently. This means they do not add any new information in solving the system of equations. In simpler terms, they are equivalent despite looking distinct at first glance.

For instance, in the given problem, we determined the equations to be dependent. Both equations after substitution simplify to the equation:\[ 2y + z = 3 \]

This outcome implies that instead of three equations contributing unique intersections, at least two just overlap as the same line. Recognizing dependent equations is crucial because it reveals that the system doesn’t have uniquely determined solutions but rather an infinite set described by a parameter, here shown as \( z \).

You learn that any combination of \( y \) and \( z \) which satisfies \( 2y + z = 3 \) will also satisfy other equations if they are dependent.

Analytical Methods

Analytical methods involve using mathematical reasoning and algebraic manipulations to find solutions. These require a step-by-step approach where you solve equations using algebraic techniques such as substitution, elimination, and simplification.

In the solved system of equations, analytical method facilitates the process by first isolating a single variable from one equation, making strategic substitutions in others. This significantly simplifies solving, turning possibly complex systems into basic linear equations.

• Substitution Method: We started by isolating \( x \) in the equation \( x - y - z = 0 \) to get \( x = y + z \). This makes it easier to swap variables into other equations.
• Simplification: After substitution, simplifying the terms to dissect the system further. Here simplification revealed the dependent nature of equations.

These steps collectively quickly lead us to realize there are overlapping solutions. This insight based on systematic reasoning can be powerful when tackling complex systems.

Linear Algebra

Linear algebra is the branch of mathematics dealing with vectors, vector spaces, and linear equations like those in our problem. Systems of equations, like the one we solved, are neatly encapsulated in linear algebra terms and employed to model consistent and multi-dimensional data.

When we

• Understand Vector Relationships: Each equation aligns with a hyperplane in multi-dimensional space. Linear algebra allows systems to be solved using matrices, making multidimensional visualization possible.
• Infinite Solutions: When dependent, solutions are often on the intersection line of overlapping hyperplanes. This insight from linear algebra is profound, showing how systems behave in space.

By leveraging matrix theories and linear algebra properties, you'd easily see the structure behind what might seem like mere numbers and equations. These deep frameworks simplify finding solutions and understanding behaviors of such systems in contexts way beyond mere pencil-and-paper arithmetic.