Problem 38 - A Graphical Approach to College Algebra

Question

\(\begin{aligned} 2 x+y+3 z &=4 \\ -3 x-y-4 z &=5 \\ x+y+2 z &=0 \end{aligned}\)

Step-by-Step Solution

Answer

The system of equations is inconsistent.

Step 1: Write the System of Equations

First, let's clearly write down the system of equations we want to solve: \[\begin{align*} 2x + y + 3z &= 4 \ -3x - y - 4z &= 5 \ x + y + 2z &= 0 \end{align*}\]This is a system of three linear equations with three variables: \(x\), \(y\), and \(z\).

Step 2: Eliminate y from Equations 1 and 2

To simplify the system, we will eliminate \(y\) from equations (1) and (2). Add equation (1) and equation (2) directly:\[(2x + y + 3z) + (-3x - y - 4z) = 4 + 5\]This simplifies to:\[-x - z = 9\] (Equation 4)

Step 3: Eliminate y from Equations 1 and 3

Next, eliminate \(y\) using equations (1) and (3). Subtract equation (3) from equation (1):\[(2x + y + 3z) - (x + y + 2z) = 4 - 0\]This simplifies to:\[x + z = 4\] (Equation 5)

Step 4: Solve for z using Equations 4 and 5

Use equations (4) and (5) to solve for \(z\). Add equations (4) and (5):\[-x - z + x + z = 9 + 4\]This simplifies to:\[0 = 13\] We realize a mistake has been made; correct the subtraction in equation derivation:Instead, from (4) and (5), solve the system:\[-x - z = 9 \] \[x + z = 4 \]Add these, and solve:\[-x - z + x + z = 9 + 4 \rightarrow 0 = 13 \] Since this calculation gives an inconsistency, the original intention to solve must be reviewed.

Key Concepts

Elimination Method

The elimination method is a powerful technique used to solve systems of linear equations. The main idea is to eliminate one variable by adding or subtracting equations. This simplifies the system, making it easier to solve.

• First, you choose which variable to eliminate. In our scenario, we chose to eliminate \(y\) from the equations.
• The goal is to create equations that do not have this variable, reducing the number of variables in the system.
• Often, this involves combining equations in such a way that the selected variable is canceled out.

For example, by adding equations (1) and (2) in our exercise, the variable \(y\) is effectively eliminated, which simplifies the problem to an equation with two variables: \(-x - z = 9\). This shrinks the complexity and ultimately leads you closer to finding the solution.

Three Variables

A system with three variables, such as \(x\), \(y\), and \(z\), is common in algebra and represents a more complex challenge than systems with just two variables.

• Each equation represents a plane in three-dimensional space, and the solution to the system is the intersection point of these planes.
• A solution exists if all planes intersect at a single point.
• However, if adjustments in the arithmetic lead to inconsistency, as seen with \(0 = 13\), it might indicate the absence of a unique solution.

Systems with three variables often require strategic manipulation of equations, such as the elimination method, to simplify the system into fewer unknowns and thus, into an easily solvable form.

Algebraic Manipulation

Algebraic manipulation is a key skill used to transform equations into simpler forms, making it easier to solve them.

• It involves basic operations like addition, subtraction, multiplication, and division applied strategically to isolate variables.
• For example, subtracting equation (3) from equation (1) helped remove \(y\) in our given exercise.
• Correct manipulation of these equations helps highlight potential errors or confirm consistency within the system, as an incorrect calculation can lead to impossible results, such as \(0 = 13\).

Being meticulous with algebraic manipulation is key, as errors in these steps can either mask or incorrectly reveal solutions. Always re-examine each algebraic step if results seem inconsistent or impossible.