Problem 17 - Algebra 1: Concepts and Skills

Question

Use linear combinations to solve the linear system. Then check your solution. \(2 g-3 h=0\) \(3 g-2 h=5\)

Step-by-Step Solution

Answer

The system of equations has no solution.

Step 1: Arrange the equations

The given equations are: \(2g - 3h = 0\) and \(3g - 2h = 5\). The first equation can be rearranged as \(2g = 3h\), and the second one remains the same.

Step 2: Replace in the second equation

Substitute \(2g = 3h\) into the second equation replacing \(g\). This will yield a new equation: \(3*(2h/3) - 2h = 5\), which simplifies to \(2h - 2h = 5\). This obviously leads to a false statement, 0=5.

Step 3: Checking inconsistent solution

Since the operation provided a false statement, it can be determined that the system of equations has no solution or it is inconsistent.

Key Concepts

Linear System

A linear system is a collection of two or more linear equations involving the same set of variables. In the context of this exercise, the variables are \(g\) and \(h\). Each equation in the system represents a line in a two-dimensional space.
For linear systems, we often try to find a point where all these lines intersect, which would be the solution to the system. In simpler terms, we are looking for a set of values for the variables that satisfy all the given equations simultaneously.
We solve these systems using various methods like substitution, elimination, or matrix operations. In our given exercise, the linear system consists of two equations:

• \(2g - 3h = 0\)
• \(3g - 2h = 5\)

Using techniques like linear combinations, we try to manipulate these equations to find the variable values. However, as we will see, sometimes these systems can have peculiar outcomes.

Inconsistent Equations

Inconsistent equations are equations in a linear system that do not have any common solution. This means there's no single value for the variables that will satisfy all equations simultaneously.
When solving the exercise, we use substitution by expressing one variable in terms of the other and plugging it into the remaining equation. This process is known as forming a linear combination.
For the problem at hand:

• We rearrange the first equation: \(2g = 3h\)
• Substitute it into the second equation, resulting in a simplified equation: \(2h - 2h = 5\)

The result \(0 = 5\) is clearly false, indicating inconsistency.
Inconsistent systems like this occur when the lines representing the equations are parallel but not overlapping. In such cases, there is no solution, as there are no points common to all lines.

Checking Solutions

Checking solutions is a crucial step in solving linear systems, as it ensures the accuracy and validity of the solution. After manipulating the equations, we need to verify if our findings are correct.
In this exercise, the suggested solution process results in an inconsistency, making it essential to check the work.
Upon encountering the false statement \(0 = 5\), we can conclude there is no solution to the system.
Steps for checking solutions generally include:

• Plugging the values back into the original equations to verify they hold true.
• Ensuring there are no arithmetic errors in the derived step.
• Concluding whether the system is consistent, inconsistent, or might have infinitely many solutions.

By diligently checking our work, we can confidently identify whether the equations are truly incompatible, consistent, or otherwise.