Problem 44 - Intermediate Algebra

Question

Solve each system, if possible. If a system is inconsistent or if the equations are dependent, state this. $$ \left\\{\begin{array}{l} 0.02 a=0.02-0.03 b-0.01 c \\ 4 a+6 b+2 c-5=0 \\ a+c=3+2 b \end{array}\right. $$

Step-by-Step Solution

Answer

The system is inconsistent, with no solutions.

Step 1: Rearrange Equation 1

Firstly, let's rearrange the first equation by isolating one of the variables. The equation is given by: \[ 0.02a = 0.02 - 0.03b - 0.01c \] To isolate \(a\), divide each term by 0.02:\[ a = 1.0 - 1.5b - 0.5c \]

Step 2: Substitute Equation 1 into Equation 2

Next, substitute the expression for \(a\) from Step 1 into the second equation:\[ 4a + 6b + 2c - 5 = 0 \]Substitute \(a = 1.0 - 1.5b - 0.5c\):\[ 4(1.0 - 1.5b - 0.5c) + 6b + 2c - 5 = 0 \]Simplify:\[ 4 - 6b - 2c + 6b + 2c - 5 = 0 \]\[ -1 = 0 \]This simplifies to a false statement, indicating an inconsistency.

Step 3: Conclude the System is Inconsistent

Since the substitution leads to a false statement \(-1 = 0\), it shows that the system is inconsistent. This means there is no common solution for the given equations.

Key Concepts

Inconsistent Systems

In the context of systems of equations, an inconsistent system is one that has no solution. This occurs because the equations in the system contradict each other in such a way that they cannot all be true simultaneously. As a result, no set of variable values can satisfy all the equations at once.
To identify an inconsistent system, look for a scenario where substituting variables leads to impossibilities, such as contradictory statements. For instance, in this exercise, the process of substituting variables led to the false statement \(-1 = 0\). Such outcomes are clear indicators of an inconsistency.
Recognizing inconsistent systems is important:

• It saves time by indicating when further solving attempts are futile as no solution exists.
• It helps understand the nature of the relationships described by the equations.

Overall, determining inconsistency early in your solution process can guide you effectively toward understanding the behavior of the given equations.

Substitution Method

The substitution method is a powerful technique used to solve systems of equations. It involves solving one of the equations for one variable and then substituting that expression into the other equations. This reduces the number of variables and equations by one, making the system simpler to solve.
Here are the basic steps often followed in the substitution method:

• Solve one equation for one variable.
• Substitute this expression into the other equations.
• Solve the resulting equation, which should now contain fewer variables.

In this exercise, the substitution method revealed an inconsistency after substituting the expression for \(a\) back into the second equation, leading to an impossibility. This clear outcome demonstrated the presence of an inconsistency, helping us understand that the system can't be resolved patiently through typical means.
This demonstration shows why the substitution method is exceedingly useful, as it directly pointed to the contradiction within the system.

Dependent Equations

Dependent equations refer to equations in a system that are essentially multiples or linear combinations of one another. When this occurs, each equation does not provide additional information or constraints on the solution, indicating that they describe the same line or plane.
While not directly applicable to the solution of this problem due to inconsistency, understanding dependent equations is important because:

• If equations are dependent without contradiction, the system has infinitely many solutions, forming a line or plane of solutions.
• It highlights redundancy within the system, as not all equations contribute uniquely to finding solutions.

To check for dependency, transform the equations and check if you can express one as a scalar multiple or linear combination of others. Recognizing dependent equations helps in identifying the nature of the system and can determine if solutions are possible or redundant.