Problem 33 - Understanding Elementary Algebra with Geometry

Question

In Exercises \(29-62,\) solve the system of equations using any method you choose. $$\left\\{\begin{aligned} 8 a+6 b &=-3 \\ 12 a+9 b &=-5 \end{aligned}\right.$$

Step-by-Step Solution

Answer

The system of equations has no solution because it is inconsistent.

Step 1: Identify the equations

The system of equations given is: 1. \(8a + 6b = -3\) 2. \(12a + 9b = -5\).

Step 2: Simplify the equations

Notice that both equations can be simplified. To simplify the first equation, divide everything by 2: \(4a + 3b = -\frac{3}{2}\) The second equation can be simplified by dividing everything by 3: \(4a + 3b = -\frac{5}{3}\)

Step 3: Compare the simplified equations

After simplification both equations are: \(4a + 3b = -\frac{3}{2}\) \(4a + 3b = -\frac{5}{3}\)

Step 4: Analyze the simplified system

Both equations have the same left side but different right sides. \(4a + 3b\) cannot simultaneously be \(-\frac{3}{2}\) and \(-\frac{5}{3}\). Therefore, these equations are inconsistent and the system of equations has no solution.

Key Concepts

Inconsistent Systems

An inconsistent system of equations is one where the equations contradict each other.
This means there is no solution that satisfies all equations simultaneously.
In the provided exercise:

• We simplified the system to get two equations: \(4a + 3b = -\frac{3}{2}\) and \(4a + 3b = -\frac{5}{3}\).
• Notice how the left sides of the equations are identical: \(4a + 3b\).
• However, the right sides are different: \(-\frac{3}{2}\) and \(-\frac{5}{3}\).

This means there's no possible combination of \(a\) and \(b\) that can make these two equations true at the same time.
So, we conclude that the system is inconsistent and has no solution.
Learning to recognize inconsistent systems is crucial since it saves you time by identifying systems that have no possible solutions.

Equation Simplification

Simplifying equations makes systems easier to analyze and solve.
In this exercise, we simplified the given equations:

• The first equation: \(8a + 6b = -3\) was divided by 2, resulting in \(4a + 3b = -\frac{3}{2}\).
• The second equation: \(12a + 9b = -5\) was divided by 3, giving \(4a + 3b = -\frac{5}{3}\).

This simplification process is crucial because:

• It makes the equations easier to compare.
• It reveals any inconsistencies or dependencies between the equations more clearly.

Always consider simplifying your equations first, as it might save you time and reveal the nature of the solution early on.

Methods for Solving Systems

Several methods exist to solve linear systems of equations. Here are some common ones:

• Substitution Method:
  Solve one equation for one variable, then substitute that expression into the other equation.
• Elimination Method:
  Add or subtract equations to eliminate one variable, making it easier to solve for the remaining variable.
• Graphing:
  Graph each equation on the same set of axes and find the intersection point. This point is the solution to the system.

In our example, we used equation simplification to quickly identify the system as inconsistent.
Recognizing which method to use can depend on the specific system you are dealing with.
Sometimes a combination of methods works best.
It's essential to be familiar with all these methods to choose the best approach for different types of systems.