Solve the system.
[tex]\begin{aligned}
-3 x+3 y+3 z & =3 \
-5 x-5 y+z & =-3 \
15 x+15 y-3 z & =8
\end{aligned}[/tex]
Answer (2)
Simplify the first equation: − x + y + z = 1 .
Multiply the simplified equation by 3: − 3 x + 3 y + 3 z = 3 .
Eliminate z from the third equation, resulting in 0 = − 1 , which is a contradiction.
Conclude that the system has no solution and is inconsistent: The system has no solution. The system is inconsistent.
Explanation
Analyzing the Problem We are given a system of three linear equations with three unknowns: x, y, and z. The equations are:
-3x + 3y + 3z = 3 -5x - 5y + z = -3 15x + 15y - 3z = 8
We need to solve this system and determine if it has a unique solution, no solution (inconsistent), or infinitely many solutions (dependent).
Simplifying the First Equation First, let's simplify the first equation by dividing by 3:
-x + y + z = 1
Multiplying the Simplified Equation by 3 Now, let's multiply the simplified first equation by 3:
-3x + 3y + 3z = 3
Eliminating z from the Third Equation Next, we'll add 3 times the second equation to the third equation to eliminate z from the third equation. This gives:
3*(-5x - 5y + z) + (15x + 15y - 3z) = 3*(-3) + 8
Which simplifies to:
-15x - 15y + 3z + 15x + 15y - 3z = -9 + 8
0 = -1
This indicates an inconsistency in the system.
Determining the Nature of the Solution Since we arrived at a contradiction (0 = -1), the system has no solution. Therefore, the system is inconsistent.
Examples
Systems of equations are used in various fields, such as economics, engineering, and computer science. For example, in economics, systems of equations can be used to model the supply and demand of goods in a market. In engineering, they can be used to analyze the forces acting on a structure. In computer science, they can be used to solve problems in areas such as linear programming and network analysis. Understanding how to solve systems of equations is a fundamental skill in these fields.
The system of equations has no solution since manipulating the equations leads to a contradiction (9 = 8). Therefore, it is inconsistent. The overall conclusion is that there is no solution to the system.
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