Problem 22 - College Algebra with Modeling and Visualization

Question

If possible, solve the system. $$ \begin{aligned} a-2 b+c &=-1 \\ a+5 b &=-3 \\ 2 a+3 b+c &=-2 \end{aligned} $$

Step-by-Step Solution

Answer

The system has no solution.

Step 1: Identify the System of Equations

We have the following system of equations: 1. \( a - 2b + c = -1 \) 2. \( a + 5b = -3 \) 3. \( 2a + 3b + c = -2 \). Our goal is to find the values of \( a \), \( b \), and \( c \) that satisfy all three equations.

Step 2: Simplify by Substitution

From Equation 2, \( a + 5b = -3 \), we can solve for \( a \): \( a = -3 - 5b \). We'll substitute this expression for \( a \) into Equations 1 and 3 to eliminate \( a \).

Step 3: Substitute in Equation 1

Substitute \( a = -3 - 5b \) into Equation 1: \[ (-3 - 5b) - 2b + c = -1 \]Simplify:\[ -3 - 7b + c = -1 \]Rearrange to find \( c \):\[ c = 7b - 2 \]

Step 4: Substitute in Equation 3

Substitute \( a = -3 - 5b \) into Equation 3: \[ 2(-3 - 5b) + 3b + c = -2 \]This becomes:\[ -6 - 10b + 3b + c = -2 \]Simplify:\[ -7b + c = 4 \]Using \( c = 7b - 2 \) from Step 3, substitute for \( c \): \[ -7b + (7b - 2) = 4 \]This simplifies to:\[ -2 = 4 \]

Step 5: Analyze the Result

The equation \( -2 = 4 \) is a contradiction, which means there is no set of values for \( a \), \( b \), and \( c \) that can satisfy all three equations simultaneously. This system of equations has no solution.

Key Concepts

Substitution Method

The substitution method is a powerful tool for solving systems of linear equations. You start by solving one of the equations for one variable. Then, substitute that expression into the other equations. This reduces the number of variables and helps in isolating a solution.
For the given system, the second equation, \( a + 5b = -3 \), is used. We solve for \( a \) to get \( a = -3 - 5b \).
Substituting this expression into the other two equations helps in eliminating \( a \), making it easier to manage and reduce the system to terms of \( b \) and \( c \).

• This method works well when one equation is easy to solve for a variable.
• It systematically eliminates variables to solve the system step-by-step.

Substituting can often reveal insightful forms of combining equations, potentially leading to solutions that might not be immediately obvious from the original system.

Contradiction in Equations

A contradiction in equations arises when simplifying a system leads to an impossible statement. In the exercise, we arrive at the sentence \(-2 = 4\). This indicates a contradiction, as it is clearly untrue.
This outcome suggests that no solutions exist that can satisfy all given equations simultaneously.
When such contradictions occur, it means that the set of equations represents geometrically parallel lines, leading to no points of intersection.

• While solving, encountering such contradictions indicate that the system is inconsistent.
• It is helpful to verify each step during simplification to confirm the system's logical consistency.

Identifying contradictions is crucial as it saves time by avoiding endless calculations in search of non-existent solutions.

No Solution in Systems of Equations

When we say a system of equations has no solution, it implies an inconsistency in the overall set of equations. The equations do not intersect at any common point:
They are parallel and distinct in geometric terms.

In the context of the given problem, the inconsistency arose from substituting and simplifying the equations until reaching a false statement. This signals parallel planes or lines in their graphical representation.

• No solution means the graphical representation of the system shows no common intersection point.
• Identifying this early saves computational resources and time.

Understanding when and why systems have no solutions is quite valuable in mathematical analysis and applied scenarios, such as in linear programming or network flows.