Problem 38 - Intermediate Algebra

Question

Solve each system. If a system is inconsistent or if the equations are dependent, state this. $$ \left\\{\begin{array}{l} 2 a-b+c=6 \\ -5 a-2 b-4 c=-30 \\ a+b+c=8 \end{array}\right. $$

Step-by-Step Solution

Answer

The system is dependent with infinitely many solutions: \((a, b, c) = (2 + 2b, b, 10 - 3b)\).

Step 1: Examine the System of Equations

We have the following system of equations: \[ \begin{align*}1) & \, 2a - b + c = 6 \2) & \, -5a - 2b - 4c = -30 \3) & \, a + b + c = 8\end{align*} \ \] Our task is to solve this system for \(a\), \(b\), and \(c\).

Step 2: Use Substitution or Elimination

We will use elimination to reduce the system. Start by eliminating \(a\). From equation (3), rewrite the expression: \[ a = 8 - b - c \]Substitute this expression for \(a\) in both equations (1) and (2).

Step 3: Substitute for a in Equations

Replace \(a\) in equation (1):\[2(8 - b - c) - b + c = 6 \] Which simplifies to: \[16 - 2b - 2c - b + c = 6 \] Which simplifies further to: \[16 - 3b - c = 6 \] Thus:\[3b + c = 10 ag{4} \] Now replace \(a\) in equation (2):\[-5(8 - b - c) - 2b - 4c = -30\] Which simplifies to:\[-40 + 5b + 5c - 2b - 4c = -30\] Thus:\[ 3b + c = 10 ag{5}\]

Step 4: Compare Results from Reduction

Notice that equations (4) and (5) are identical, leading both to the equation: \[ 3b + c = 10 \] This indicates equation (1) and equation (2) simplified to the same result when \(a\) was replaced.

Step 5: Analyze Dependency of Equations

Upon solving for \(b\) and \(c\), we see that equation (1) and (2) provided no new information beyond the relation \(3b + c = 10\). Solve equation (3) by: \[ a + b + c = 8 ag{3} \] This already expresses one form of a line in 3D, showing dependence in the linear combinations of the equations. The system is dependent and has infinitely many solutions.

Step 6: Express the Solution

We will express the solution in terms of \(b\) because of the dependency. Using equation \(3b + c = 10\), express:\[ c = 10 - 3b \] Then substitute into equation (3) to find \(a\): \[ a = 8 - b - (10 - 3b) \ = 8 - b - 10 + 3b \ = 2 + 2b \] Thus, the solution parameters are: \[ a = 2 + 2b, \quad b = b, \quad c = 10 - 3b \]

Step 7: State the General Solution

The solution set for the system is given by: \[ (a, b, c) = (2 + 2b, b, 10 - 3b) \] This implies that there are infinitely many solutions along the line defined by these parameters.

Key Concepts

Dependent System

When a system of linear equations is considered 'dependent,' it means that at least one of the equations in the system can be derived from the others. Essentially, one equation is a linear combination of others, indicating their lack of independence. This lack of independence manifests because the equations describe the same geometric entity, like a line or a plane in a three-dimensional space. In the context of our example, we examined three equations that seemed distinct at first glance.
Upon applying the elimination method to the system:

• The process revealed that two equations simplify to give the same equation: \(3b + c = 10\).
• This simplification showed that these two, originally different equations, are dependent.

When you encounter a dependent system, you'll find that the system cannot be easily solved for a unique solution. Instead, they reveal a plane or line where solutions are found, instead of a single point in space.

Infinitely Many Solutions

A system of equations with 'infinitely many solutions' is one where the same relationship holds true for multiple possible values of the variables involved. This often occurs when the equations represent the same line or plane in a coordinate system, offering multiple intersection points.
For the provided system of equations:

• The dependent equations resulted in simplifying to \(3b + c = 10\), showing multiple solutions exist along a line.
• Using substitution, we found expressions for \(a\), \(b\), and \(c\) that depend on a parameter \(b\).
• These expressions indicate that for any real value of \(b\), there corresponds a pair \(a, c\) that satisfies the entire system.

Thus, in this situation, rather than pinpointing a single solution, the solutions describe a line of points, indicating an unlimited number of solutions where the variables can adjust but still satisfy all original equations.

Elimination Method

The 'elimination method' is a systematic approach used to solve systems of equations by eliminating one variable at a time. This method transforms the original system into a simpler form, often revealing the relationships between variables more clearly.
To implement the elimination method in solving our example:

• We decided to first eliminate the variable \(a\) by substituting an expression for \(a\) derived from one of the equations into the others.
• By careful substitution, and simplifying both equations, we reduced the system into a single equation involving \(b\) and \(c\), \(3b + c = 10\).
• With this equation, we then substituted it back into another equation to express \(a\) in terms of \(b\).

This method is powerful because it allows us to simplify complications and dependencies in the system, bringing patterns or dependencies into focus. It’s effective not only for solving but also for identifying when equations might be dependent or when the system has infinitely many solutions or none at all.