Problem 49 - Intermediate Algebra

Question

Solve each system, if possible. If a system is inconsistent or if the equations are dependent, state this. $$ \left\\{\begin{array}{l} r+s+4 t=3 \\ 3 r+7 t=0 \\ 3 s+5 t=0 \end{array}\right. $$

Step-by-Step Solution

Answer

The system is inconsistent, as shown by the contradiction 0 = 3.

Step 1: Set Up the System of Equations

We start by writing down the system of equations as follows: 1. \( r + s + 4t = 3 \)2. \( 3r + 7t = 0 \)3. \( 3s + 5t = 0 \).

Step 2: Solve Equation for r

Focus on equation 2, \( 3r + 7t = 0 \). Solve for \( r \):\[ r = -\frac{7}{3}t \].

Step 3: Solve Equation for s

Now consider equation 3, \( 3s + 5t = 0 \). Solve for \( s \):\[ s = -\frac{5}{3}t \].

Step 4: Substitute r and s into the First Equation

Substitute \( r = -\frac{7}{3}t \) and \( s = -\frac{5}{3}t \) into equation 1:\[ -\frac{7}{3}t - \frac{5}{3}t + 4t = 3 \].

Step 5: Simplify and Solve for t

Combine like terms in equation 1 substitution:\[ -4t + 4t = 3 \].This simplifies to 0 = 3, which is a contradiction.

Step 6: Analyze the Result

The equation 0 = 3 is a contradiction, showing that there is no consistent solution to this system. Hence, the system of equations is inconsistent.

Key Concepts

Inconsistent Systems

An inconsistent system of equations occurs when there is no solution that satisfies all the equations simultaneously. This means the equations in the system are contradictory. In our example, after substituting values into the first equation, the statement simplified to \( 0 = 3 \). This is a clear contradiction since 0 is never equal to 3. Thus, there are no values of \( r \), \( s \), or \( t \) that can satisfy all the given equations at the same time.

• Inconsistent systems do not intersect at any point.
• In terms of graphing, their lines or planes do not meet.

This is important in understanding real-life problems where certain requirements cannot be simultaneously fulfilled within given constraints.

Dependent Equations

Dependent equations are essentially the same line or plane, expressed in different forms. If you can manipulate one equation into another by simple arithmetic (such as multiplication or addition), the equations are dependent.
In cases with a "dependent" system, one equation can be derived from another, leading to infinitely many solutions. However, in our exercise, the system was inconsistent, not dependent, because we reached a logical contradiction while trying to solve it. Thus, the equations were independent in terms of this inconsistency.

• Dependent equations share all solutions meaning they represent the same geometric figure.
• If you graph them, the lines overlap entirely.

In essence, understanding the nature of equation dependency helps predict the number of solutions a system can have.

System of Equations

A system of equations is a set of two or more equations with the same variables. The goal is to find a common solution that satisfies all equations at once. This concept is foundational in algebra and applicable in various fields such as economics, engineering, and physics.

• Each equation in the system provides a different constraint for the variables.
• Solutions are the points where all the equations intersect.

In our given system, the variables were \( r, s, \) and \( t \), and the equations created constraints between these variables. Understanding how to set up a system of equations and identify its nature (consistent, inconsistent, or dependent) is key to solving algebraic problems.

Solving Systems of Linear Equations

Solving systems of linear equations involves finding the values for the variables that satisfy all provided equations. Depending on the equations' nature, a system can have a single solution, no solution, or infinitely many solutions.
In our step-by-step exercise, we employed substitution to express \( r \) and \( s \) in terms of \( t \). By plugging these into the first equation, we attempted to solve the system. Finding a contradiction indicated no possible solution.

• Common methods for solving include graphing, substitution, and elimination.
• Recognizing when to use each method can simplify the problem-solving process.

This process teaches problem-solving and analytical thinking, as well as the ability to interpret solutions in real-world contexts.