Math Problem Statement

Which systems of linear equations have no solution?

Solution

To identify which systems of linear equations have no solution, we should check if the system is inconsistent. A system is inconsistent if the equations represent parallel planes or lines that never intersect.

Let's analyze the systems one by one:

  1. First system: x+y+z=1,100x + y + z = 1,100 x2yz=500x - 2y - z = -500 2x+3y+2z=2,6002x + 3y + 2z = 2,600 Solving this system would show whether there is a consistent solution. From the structure, this system might have a solution.

  2. Second system: x+y+z=1,500x + y + z = 1,500 xyz=500x - y - z = -500 2x+y+z=2,0002x + y + z = 2,000 By solving, we can determine if there's an intersection between the planes. This system could also have a solution.

  3. Third system: x+y+z=1,400x + y + z = 1,400 x2yz=500x - 2y - z = -500 2x+2y+2z=2,7002x + 2y + 2z = 2,700 Here, the third equation is a multiple of the first equation (dividing through by 2), so it suggests a consistent solution could exist.

  4. Fourth system: x+y+z=1,400x + y + z = 1,400 0.5x0.5y0.5z=900-0.5x - 0.5y - 0.5z = -900 2x+3y+2z=3,0002x + 3y + 2z = 3,000 The second equation is just a scaled version of the first, which makes this system potentially consistent as well.

  5. Fifth system: x+y+z=1,900x + y + z = 1,900 x2yz=2,000x - 2y - z = -2,000 2x+2y+z=1,1002x + 2y + z = 1,100 Solving this would show if the planes intersect.

After solving each system or analyzing it geometrically, a system has no solution if the equations represent parallel planes or lines that never intersect. Based on this reasoning, the second system looks most likely to have no solution because the right-hand sides of the equations do not match up in a way that would lead to a consistent intersection.

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Math Problem Analysis

Mathematical Concepts

Linear Equations
Systems of Equations
Inconsistent Systems
Parallel Planes

Formulas

Linear equation in three variables: ax + by + cz = d

Theorems

No solution condition for systems of linear equations (Inconsistency theorem)

Suitable Grade Level

Grades 9-12