Problem 44 - Precalculus: Mathematics for Calculus

Question

Determine whether the system of linear equations is inconsistent or dependent. If it is dependent, find the complete solution. $$\left\\{\begin{array}{rr} -2 x+6 y-2 z= & -12 \\ x-3 y+2 z= & 10 \\ -x+3 y+2 z= & 6 \end{array}\right.$$

Step-by-Step Solution

Answer

The system is dependent with solution \((x, y, z) = (3t + 2, t, 4)\) where \(t\) is any real number.

Step 1: Write the System of Equations

The system of linear equations is given as: 1. \(-2x + 6y - 2z = -12\) 2. \(x - 3y + 2z = 10\) 3. \(-x + 3y + 2z = 6\). We need to determine if the system is inconsistent (no solution) or dependent (infinitely many solutions).

Step 2: Simplify the System

Start by simplifying the equations that can lead to revealing dependency or inconsistency. Add equation 2 and equation 3 to see if they simplify more:\[ (x - 3y + 2z) + (-x + 3y + 2z) = 10 + 6 \] This simplifies to: \[ 4z = 16 \] Solving gives \(z = 4\).

Step 3: Substitute z = 4 into Other Equations

Substitute \(z = 4\) back into the original equations to see if they'll provide consistent values for \(x\) and \(y\).Substituting into equation 2:\[ x - 3y + 2(4) = 10 \] \[ x - 3y + 8 = 10 \] \[ x - 3y = 2 \] \Substituting into equation 3:\[ -x + 3y + 2(4) = 6 \] \[ -x + 3y + 8 = 6 \] \[ -x + 3y = -2 \].

Step 4: Solve for x and y

From Step 3, we have two equations:\(x - 3y = 2\) and \(-x + 3y = -2\).Add these equations:\((x - 3y) + (-x + 3y) = 2 - 2 \)\[ 0 = 0 \] This result confirms the equations were dependent (identical when adjusted) with no unique solution.

Step 5: General Solution for Dependent System

Since we have established that the system is dependent, find a parametric solution by solving one variable in terms of others.Taking \(x = 3y + 2\), substitute \(y = t\):- \(x = 3t + 2\)- \(z = 4\)The general solution is:\((x, y, z) = (3t + 2, t, 4)\) where \( t \) is a parameter.

Key Concepts

Inconsistent Systems

A system of linear equations is known as inconsistent if it has no solution. This means the lines represented by the equations do not intersect at any point in space. The equations contradict each other, resulting in no common solution that can satisfy all the equations simultaneously. If you ever encounter a situation with equations where, after simplification, you arrive at an absurd statement like \(1=0\), you'll know the system is inconsistent.

Trying to solve an inconsistent system can reveal underlying contradictions. For example, equations might appear parallel upon graphical representation, indicating they never meet. In inconsistent systems, each equation tends to pull in a different direction, making it impossible to satisfy them all at the same time.

Dependent Systems

Dependent systems are quite the opposite of inconsistent ones. They have an infinite number of solutions instead of none. This happens because the equations describe the same line or plane, essentially overlapping. When system equations reduce to multiple forms of the same equation, or to forms offering no new information (like \(0=0\)), the system is dependent.

Dependent systems are interesting because, while they do not provide a single, unique solution, they give us a set of equations that are layered atop one another, threading through the same path. This shows the solutions are interdependent on each parameter involved. In the step-by-step solution, combining equations revealed this dependency, showcasing how equations can essentially mirror each other.

Parametric Solutions

When dealing with dependent systems, parametric solutions are an effective way to express the infinite set of possible solutions. Instead of providing a single answer, we're able to express the solution with parameters. This introduces a variable, often denoted as \(t\) or \(s\), which can take an infinite number of values, generating numerous solutions.

In our example, once it was determined the system was dependent, we found a parametric solution by expressing \(x\) in terms of \(y\), and utilizing a parameter \(t\). This gave us region over the solutions expressed as \((x, y, z) = (3t + 2, t, 4)\). The parameter offers flexibility to understand solutions on a continuum, providing a clearer picture of possible results along an entire line or plane than just one specific point.