Problem 43 - Precalculus Mathematics for Calculus

Question

Solve the system of linear equations. $$\left\\{\begin{array}{rr} x+2 y-3 z= & -5 \\ -2 x-4 y-6 z= & 10 \\ 3 x+7 y-2 z= & -13 \end{array}\right.$$

Step-by-Step Solution

Answer

The system is inconsistent and has no solutions.

Step 1: Analyze the System

We need to solve the system of equations for the variables \(x\), \(y\), and \(z\): \(x + 2y - 3z = -5\), \(-2x - 4y - 6z = 10\), \(3x + 7y - 2z = -13\). Notice that the second equation is a multiple of the first equation.

Step 2: Simplify the System

Divide the second equation \(-2x - 4y - 6z = 10\) by \(-2\). This simplifies to: \(x + 2y + 3z = -5\). It is clear now that this equation is inconsistent with the first equation.

Step 3: Determine Inconsistency

Since equations \(1\) and \(2\) simplify to \(x + 2y - 3z = -5\) and \(x + 2y + 3z = -5\) respectively, they contradict each other (as one implies \(z = -3\), the other implies \(z = 3\)), hence the system has no solutions.

Key Concepts

Inconsistent Systems

Inconsistent systems of equations are those where no set of values can simultaneously satisfy all equations involved. This typically occurs when two equations contradict each other, meaning they represent parallel lines that never intersect. For example, consider the equations in the system provided:

• One equation simplifies to \( x + 2y - 3z = -5 \)
• Another simplifies to \( x + 2y + 3z = -5 \)

These two equations give conflicting information about the value of \( z \)—one implies a positive value while the other implies a negative value.
This contradiction makes it impossible to find a solution for the system, thus labeling the system as inconsistent.
This concept is critical as it highlights the importance of checking for redundancy or contradictions among equations.

Linear Equations

Linear equations form the backbone of many algebraic systems, consisting of variables that are neither squared nor under a square root, and are typically graphed as straight lines. They generally take the form:\[ax + by + cz = d\]where \( a \), \( b \), and \( c \) are coefficients,
and \( d \) is the constant term.
For instance, the system of equations from the exercise:
\[x + 2y - 3z = -5\] is a linear equation.
Linear equations can describe real-world relationships between quantities, allowing us to solve for unknown variables using graphical, substitution, or elimination methods.

Solving Systems of Equations

When solving systems of equations, the goal is to find values for the variables that satisfy all equations simultaneously. However, as mentioned before, some systems like inconsistent ones do not have solutions. Solving typically involves three key methods:

• Graphical Method: Plot each equation on a graph and identify the points where the lines intersect.
• Substitution Method: Solve one of the equations for a variable and substitute this into the other equations.
• Elimination Method: Add or subtract equations to eliminate one of the variables, simplifying the system step by step.

In our example, we identified that simplification led to inconsistent results, revealing no intersection point and thus no solution for the system.
Understanding these methods is essential for tackling all kinds of linear systems effectively.