Problem 106 - Algebra and Trigonometry Real Mathematics, Real People

Question

Determine whether the statement is true or false. Justify your answer. If a system of three linear equations is inconsistent, then its graph has no points common to all three equations.

Step-by-Step Solution

Answer

The given statement is true.

Step 1: Understand the definitions

In the context of system of equations, a system is referred to as inconsistent if it does not have a solution. Graphically, this means the lines representing the equations do not intersect at a common point.

Step 2: Apply the definitions to the given statement

The statement says: 'If a system of three linear equations is inconsistent, then its graph has no points common to all three equations.' If we apply our understanding of an inconsistent system, we can see that an inconsistent system would mean there are no common solutions, or graphically, no common points where all equations intersect.

Step 3: Determine the truthfulness of the statement

Based on the definitions, we can conclude that the statement is true. If a system of three linear equations is inconsistent, then indeed its graph would not have any points common to all three equations, because an inconsistent system, by definition, means there are no solutions or intersecting points for all equations.