Problem 30 - Algebra 2

Question

Melissa is solving the system of equations \(r+2 s+t=3\), \(r+2 s+t=3\), \(2 r+4 s+2 t=6,\) and \(3 r+6 s+3 t=12 .\) Is she correct? Explain.

Step-by-Step Solution

Answer

Melissa is correct; the equations are dependent, representing the same plane.

Step 1: Write the Given Equations

The given system of equations is: 1. \( r + 2s + t = 3 \) 2. \( r + 2s + t = 3 \) 3. \( 2r + 4s + 2t = 6 \) 4. \( 3r + 6s + 3t = 12 \). Examine these equations.

Step 2: Compare Equations

Compare the first and second equations: \( r + 2s + t = 3 \) is repeated. The first two equations are identical.

Step 3: Simplify Equations

Notice that the third equation \( 2r + 4s + 2t = 6 \) is simply twice the first equation. Similarly, the fourth equation \( 3r + 6s + 3t = 12 \) is three times the first equation.

Step 4: Analyze Redundancy

Since each equation is a multiple of the first, they all represent the same plane in the 3D space, indicating redundancy in the system of equations.

Step 5: Determine Consistency

Since all equations represent the same plane, there are infinitely many solutions where \( r \), \( s \), and \( t \) can satisfy \( r + 2s + t = 3 \). The system is consistent but dependent, not providing a unique solution.

Key Concepts

Redundant Equations

When we talk about redundant equations in a system, we refer to equations that do not add new information to the system. In Melissa's case, we have a set of four equations:

• Firstly, observe equations 1 & 2: \( r + 2s + t = 3 \). These two are exactly the same, exhibiting redundancy.
• Then, let's take a look at equation 3: \( 2r + 4s + 2t = 6 \). This equation is a simple multiple of the first one (twice its value).
• Similarly, equation 4: \( 3r + 6s + 3t = 12 \) is three times the first equation.

Redundancy occurs when additional equations do not help pinpoint unique solution sets but simply rephrase or duplicate the information already given. In this situation, Melissa's system doesn’t have new restrictions since all equations describe the same condition: \( r + 2s + t = 3 \). Therefore, all represent the same fundamental relationship.

Consistent System

A system of equations is considered consistent if there is at least one set of solutions that satisfies all the equations in the system. For Melissa's system, which is composed of redundant equations, all equations coincide on the same plane in a three-dimensional space:

• Since all equations boil down to \( r + 2s + t = 3 \), there are sets of \( r \), \( s \), and \( t \) that make this statement true.
• These consistent solutions form a line or plane in the space.

Despite having three independent variables, the redundancy in the equations limits the range of possible unique solutions, resulting in a consistent system where multiple value combinations fulfill the equations. Therefore, a consistent system does not necessarily mean a single solution but means having any solutions at all without contradictions.

Infinitely Many Solutions

In the context of Melissa's system, having infinitely many solutions means there are endless combinations of \( r \), \( s \), and \( t \) that satisfy the primary equation \( r + 2s + t = 3 \). Here's how it breaks down:

• Imagine the equation as a flat plane slicing through a 3D space. Any point on this plane is a valid solution, thus creating infinitely many possibilities.
• Because Melissa's equations all describe this same plane, any change in one of the variables can be compensated by changes in others, without exiting the plane.

This situation typically contrasts with unique solutions, where the intersection of distinct equations would pin down specific values for each variable. In Melissa's case, since there is no such intersection from different equations, the solution set is infinite.

Dependent System

A dependent system occurs when the equations within the system do not provide new direction or independent conditions, meaning they rely on one another. Melissa's system is dependent because:

• Each equation reiterates the same relationship, mathematically interconnected by being multiples of each other.
• There's no unique path produced by combining the equations. They collectively define one single solution set, not multiple independent ones.

The dependency restricts the equations from establishing distinct points or lines of intersection. Thus, despite having multiple equations, the dependency reduces them down to represent the same geometrical figure (a plane in this case), indicating no unique solution emerges without variation among the equations.