Problem 39 - A Graphical Approach to College Algebra

Question

\(\begin{array}{l} x+y+z=0 \\ x-y-z=3 \\ x+3 y+3 z=5 \end{array}\)

Step-by-Step Solution

Answer

The equations suggest an inconsistency indicating no single solution in the real number set.

Step 1: Choose Two Equations to Eliminate a Variable

Let's start with the first and second equations: \[ x + y + z = 0 \] \[ x - y - z = 3 \] We'll add them to eliminate \( y \) and \( z \).

Step 2: Add Equations to Eliminate Variables

By adding the equations from Step 1, we get:\[ (x + y + z) + (x - y - z) = 0 + 3 \]This simplifies to:\[ 2x = 3 \]Dividing both sides by 2 gives:\[ x = \frac{3}{2} \]

Step 3: Substitute Back to Find y and z

Substitute \( x = \frac{3}{2} \) into the first equation:\[ \frac{3}{2} + y + z = 0 \]This gives:\[ y + z = -\frac{3}{2} \]Substitute \( x = \frac{3}{2} \) into the third equation:\[ \frac{3}{2} + 3y + 3z = 5 \]This gives:\[ 3y + 3z = \frac{7}{2} \]Divide by 3:\[ y + z = \frac{7}{6} \]

Step 4: Solve Two Simplified Equations for y and z

We now have two new equations in terms of \( y \) and \( z \):\[ y + z = -\frac{3}{2} \]\[ y + z = \frac{7}{6} \]However, these are inconsistent, indicating a mistake or that the system is over-determined. Given a possible solution, review and ensure correct steps in solving or interpreting physical scenarios correctly.

Key Concepts

Systems of Equations

Systems of equations consist of multiple equations with several variables. The primary goal is to find values for the variables that satisfy all equations simultaneously.
A system of equations is consistent if at least one set of solutions exists. Conversely, it is inconsistent if no solutions can satisfy all equations.
The given system in the exercise can be expressed as a set of three equations relating three variables:

• The first equation: \(x + y + z = 0\)
• The second equation: \(x - y - z = 3\)
• The third equation: \(x + 3y + 3z = 5\)

To find a solution, these equations can be solved together to find a common set of values for \(x\), \(y\), and \(z\). Systems of equations can be solved using various methods such as substitution, graphing, and elimination.

Elimination Method

The elimination method is a technique used to eliminate one of the variables by adding or subtracting equations. This method is helpful when equations contain similar terms or when equations can be easily manipulated to facilitate the elimination of variables.
In the exercise, the elimination method was used by adding the first equation to the second equation to remove \(y\) and \(z\):

• First equation: \(x + y + z = 0\)
• Second equation: \(x - y - z = 3\)

Adding these equations results in: \[(x + y + z) + (x - y - z) = 0 + 3\]This operation simplifies to:\[2x = 3\]
Using the elimination method combines equations wisely to isolate one variable step by step, making it easier to solve for others subsequently once one value is determined.

Variable Elimination

Variable elimination involves strategically removing a variable from a system of equations to simplify solving.
After finding \(x = \frac{3}{2}\) from the elimination method, substitute \(x\) back into other equations to reduce them to equations with only \(y\) and \(z\).

• Start by substituting into the first equation: \(\frac{3}{2} + y + z = 0\), simplifying to \(y + z = -\frac{3}{2}\)
• Next, substitute into the third equation: \(\frac{3}{2} + 3y + 3z = 5\), simplifying to \(y + z = \frac{7}{6}\)

This process aids in finding the values for other variables. However, care should be taken as simplifications can sometimes lead to inconsistencies, indicating either no solution, potential errors, or over-determined systems as highlighted in the original solution steps.