91Ó°ÊÓ

Inconsistent systems occur when there are no solutions that satisfy all equations involved. This happens because the equations in the system represent parallel lines in a geometric sense.
Imagine two lines on a graph that never meet, they are parallel. When this happens in a system of equations, we have an inconsistent system.
For instance, in the provided exercise, after substituting the value of x from the first equation into the second, we received the impossible equation: 0 = 10.
This contradiction confirms that the two lines represented by the equations never intersect, making the system inconsistent.
When working with systems of equations, always look for such contradictions to determine if the system is inconsistent.

The substitution method is a powerful technique for solving systems of linear equations. It involves expressing one variable in terms of another and then substituting this expression into the remaining equations.
Here's a simplified approach:

• Simplify one of the equations and solve for one variable in terms of the other.
• Substitute this expression into the other equation(s).
• Solve the resulting equation for the remaining variable.
• Substitute back to find the other variable.

The step-by-step example in the exercise used this method:

1. First, we simplified the first equation to express x in terms of y as: \( x = 3y \).
2. Then, we substituted \( x = 3y \) into the second equation.
3. The result was \( -21y + 21y = 10 \), leading to the contradiction \( 0 = 10 \), indicating inconsistency.

Linear equations are the foundation of algebra. They describe a straight line when graphed on a coordinate plane. A linear equation in two variables generally looks like \( ax + by = c \), where a, b, and c are constants.
Key features of linear equations include:

• They have no exponents on the variables.
• They graph as straight lines.
• The solution of a linear equation can be a single point, a line (infinitely many solutions), or no point (inconsistent system).

In solving these equations, methods like substitution, elimination, and graphical analysis are commonly used.
The exercise we analyzed showed how linear equations could reveal inconsistent systems by simplifying and substituting to see if the equations made sense together.