Problem 37 - 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 \(z\). $$\begin{aligned} 2 x+3 y+4 z &=3 \\ 6 x+3 y+8 z &=6 \\ 6 y-4 z &=1 \end{aligned}$$

Step-by-Step Solution

Answer

The equations are dependent, and the solution set is \((x, y, z) = \left(\frac{3}{4} - z, \frac{4z + 1}{6}, z\right)\)."

Step 1: Identify and Arrange the System of Equations

The given system of equations is: \[\begin{aligned} \ 2x + 3y + 4z &= 3 \ 6x + 3y + 8z &= 6 \ 6y - 4z &= 1 \end{aligned}\] We will work with these equations to determine if they are dependent and to find the solution.

Step 2: Eliminate One Variable from First Two Equations

We subtract the first equation from the second to eliminate the variable \(x\): \[\begin{aligned} &(6x + 3y + 8z) - (2x + 3y + 4z) = 6 - 3 \ &4x + 4z = 3 \ &x + z = \frac{3}{4} \end{aligned}\] This gives us the equation \(x + z = \frac{3}{4}\) as a new equation to work with.

Step 3: Solve for One Variable Explicitly

From the equation \(x + z = \frac{3}{4}\), solve for \(x\): \[x = \frac{3}{4} - z\]. Now, we have expressed \(x\) in terms of \(z\).

Step 4: Substitute into the Third Equation to Solve for y in Terms of z

Since we haven't modified the third equation \(6y - 4z = 1\), solve for \(y\): \[6y = 4z + 1 \implies y = \frac{4z + 1}{6}\]. Now, we have expressed \(y\) in terms of \(z\).

Step 5: Compile Solution Set in Terms of z

The expressions derived are \(x = \frac{3}{4} - z\) and \(y = \frac{4z + 1}{6}\). Thus, the solution set for the dependent system is given by: \[(x, y, z) = \left(\frac{3}{4} - z, \frac{4z + 1}{6}, z\right)\].

Key Concepts

Dependent System

In mathematics, a dependent system arises when the equations in a system are not independent of each other. This means that at least one of the equations in the system can be derived from the others. In the case of the exercise, the system consists of three equations, but they are not all independent. When you subtract the first equation from the second, it results in a new equation, indicating a relationship between the variables that is already implied by the original equations. This occurrence shows a dependence among them. A dependent system does not have a unique solution; instead, it has infinitely many solutions that can be written in terms of a free variable.To identify a dependent system, you look for an equation that repeats or can be generated by linearly combining other equations in the system. Here, the equations are restructured to express solutions in terms of the variable \(z\), demonstrating this dependency.

Analytical Solution

An analytical solution involves solving equations using algebraic manipulations to find the exact values of the variables. This approach gives an expression in terms of one or more variables rather than numerical approximations. For the given system, solving analytically involves performing operations like addition, subtraction, and substitution to reformulate the equations. Through steps including isolating variables and substituting expressions, you create a pathway to express solutions clearly. The analytical method provides insight and understanding of the relationships and dependencies in the system because it allows the solver to trace each connection or elimination of variables. It is valuable for checking dependency as it shows if the system can be reduced to fewer equations, thereby identifying dependent or independent nature.

Variable Elimination

Variable elimination is a method used to simplify and solve systems of equations by removing one variable at a time. In this exercise, variable elimination helped in reducing the system to deduce the dependent relationships.By strategically subtracting one equation from another, you eliminate a variable, simplifying the system to an expression with fewer variables. For example, subtracting the first equation from the second, as done in the solution, removes the \(x\) variable, reducing the complexity of the system.This method highlights the presence of dependent systems by showing how fewer equations are needed to describe the relationships among variables. After eliminating variables, you find expressions for the remaining variables in terms of one free variable \(z\), allowing you to assemble the general solution set. This approach aids in visualizing how the equations are interconnected and can simplify solving complex problems.