Problem 25 - College Algebra with Modeling and Visualization

Question

If possible, solve the system. $$ \begin{array}{l} -x+3 y+z=3 \\ 2 x+7 y+4 z=13 \\ 4 x+y+2 z=7 \end{array} $$

Step-by-Step Solution

Answer

The system has infinitely many solutions; solutions depend on values of \(y\) and \(z\).

Step 1: Write the System of Equations

We start by writing the given system of equations:\[\begin{align*}-x + 3y + z &= 3 \2x + 7y + 4z &= 13 \4x + y + 2z &= 7\end{align*}\]

Step 2: Isolate One Variable in One Equation

Let's isolate \(x\) in the first equation:\[-x + 3y + z = 3 \x = 3y + z - 3\]

Step 3: Substitute for x in the Other Equations

Substitute \(x = 3y + z - 3\) into the second and third equations:Second equation:\[2(3y + z - 3) + 7y + 4z = 13\]Simplifying gives:\[6y + 2z - 6 + 7y + 4z = 13\]\[13y + 6z = 19\]Third equation:\[4(3y + z - 3) + y + 2z = 7\]Simplifying gives:\[12y + 4z - 12 + y + 2z = 7\]\[13y + 6z = 19\]

Step 4: Check for Consistency

Notice that the simplified form of both the second and third equations is \(13y + 6z = 19\). This implies that the two equations are identical, indicating that the system may be dependent.

Step 5: Interpret the Results

Since we have an identical equation, this means the system has infinitely many solutions. The values of \(y\) and \(z\) satisfy \(13y + 6z = 19\), and \(x\) can be found using \(x = 3y + z - 3\).

Key Concepts

Dependent Equations

In linear algebra, a system of equations is said to be dependent when at least two equations represent the same geometric object, leading to redundancy in the information provided by the system. This typically occurs when one equation in the system can be expressed as a linear combination of the others.

In our solved example, the equations reduce to the same line:

• By substituting dependencies in equations, you notice that the second and third both simplify to the same equation, which illustrates they are dependent.
• This redundancy suggests a relationship where not all three original equations are necessary to define the system, as one or more provide overlapping information.

Understanding dependent equations helps us see that not all mathematical systems provide unique or additive information, which is foundational in solving and simplifying complex systems.

Infinitely Many Solutions

Infinitely many solutions occur in a system of linear equations when the system is dependent, as we've seen. Instead of intersecting at a single point, the lines or planes described by the equations coincide along a continuous line or plane of solutions.

For the example at hand:

• The identical outcomes for equations reveal a line along which the solutions lie.
• Every pair of values \(y, z\) satisfying \(13y + 6z = 19\) can yield a valid solution when substituted back to find \(x\).

This tells us that understanding infinitely many solutions allows us to appreciate the breadth of possible answers—not just a single point solution, but an entire range, providing deeper insight into the structure and behavior of the system being studied.

Linear Algebra

Linear algebra is the branch of mathematics focusing on vector spaces and linear mappings between these spaces. It forms the foundational framework for understanding systems of linear equations.

Key points related to our exercise:

• When solving the linear system, rewriting it in terms of one variable helped identify dependencies, showcasing a common technique in linear algebra.
• Understanding concepts like dependent equations and infinitely many solutions extensively uses matrix methods and vector space theory, central to linear algebra.

Linear algebra equips students with the tools to manipulate and solve complex systems, offering insights into both theoretical and practical aspects of mathematical and real-world problems. It allows for not only solving systems like the one provided but also interpreting the relationships and dimensions of the solution sets.