91Ó°ÊÓ

An inconsistent system of equations is one that has no solution. This means that there is no set of values for the variables that will satisfy all the equations in the system simultaneously.
To determine if a system is inconsistent, we can simplify the equations and compare them. If simplifying the equations leads to a contradiction (such as stating that one number equals a different number), we conclude that the system is inconsistent.
In our example, we simplified the system to the following equations:
\ (1) \( x - y = \frac{3}{5} \)
\ (2) \( x - y = 12 \)
These equations have the same coefficients for \( x \) and \( y \), but their constants are different. Since \( \frac{3}{5} \) is not equal to \( 12 \), the system is inconsistent.
Therefore, there are no values for \( x \) and \( y \) that will satisfy both equations at the same time.

A system of linear equations has infinitely many solutions when the equations describe the same line. This occurs when the equations are multiples of each other, meaning they have the same slopes and intercepts.
For instance, consider the system:
\ (1) \( 2x + 3y = 6 \)
\ (2) \( 4x + 6y = 12 \)
On simplification, we see that equation (2) is just equation (1) multiplied by 2.
To express solutions for such a system, we usually set one variable as arbitrary (let's say \( y \)) and solve for the other variable (\( x \)) in terms of the arbitrary one.
For example, from equation (1), if \( y \) is arbitrary (denoted as \( t \)), then:
\ \( 2x = 6 - 3y \)
\ \( x = 3 - \frac{3t}{2} \)
Hence, the solutions can be written as \( (3-\frac{3t}{2}, t) \), where \( t \) can be any real number.

One of the essential steps in solving systems of linear equations is simplifying them. Simplification makes it easier to observe relationships between the equations and determine solutions or inconsistencies.
Simplifying involves:

• Combining like terms
• Dividing or multiplying all terms by a common factor
• Rewriting equations in a standard form (like \( Ax + By = C \))

In the given example, the system:
\ \( 5x - 5y - 3 = 0 \)
\ \( x - y - 12 = 0 \)
After simplification becomes:
\ \( x - y = \frac{3}{5} \)
\ \( x - y = 12 \)
Here, we identified that the simplified equations have identical coefficients for \( x \) and \( y \) but different constants, highlighting an inconsistency. Proper simplification can reveal insights quickly and make solving equations more straightforward.