Question

Find the solution to the given linear system. If the system has infinite solutions, give 2 particular solutions. $$ \begin{array}{l} x_{1}+2 x_{2}=1 \\ -x_{1}-2 x_{2}=5 \end{array} $$

Short answer

The system has no solution because it results in a contradiction: \( 0 = 6 \).

Step-by-step solution

Write the System of Equations

The system of equations is given by: \( x_{1} + 2x_{2} = 1 \) and \( -x_{1} - 2x_{2} = 5 \).

Add the Equations

Add the two equations. The left sides \( (x_{1} + 2x_{2}) + (-x_{1} - 2x_{2}) \) result in \( 0 \). The right sides \( 1 + 5 \) result in \( 6 \). This gives us \( 0 = 6 \).

Interpret the Result

The equation \( 0 = 6 \) is a contradiction, indicating that the system has no solution because no numbers exist that make this statement true.

Key concepts

System of Equations

A system of equations is a collection of two or more equations with a common set of variables. In this exercise, we have two linear equations involving the variables \( x_1 \) and \( x_2 \). Each equation represents a line in a two-dimensional space. Solving the system means finding all pairs \( (x_1, x_2) \) that satisfy both equations simultaneously.

To solve such a system, we can use various methods including substitution, elimination, or graphically checking where the lines intersect. For this exercise, the elimination method is applied by adding the equations together to eliminate the variables. The expectation is often to derive a simplified equation that gives us a clearer insight into where these lines intersect (if at all).

Understanding how systems of equations work is crucial in linear algebra and many applied sciences since it helps in modeling and solving real-world problems.

Contradiction in Equations

A contradiction in equations arises when the resulting statement from our operations is logically impossible. In this case, adding the two given equations leads to the statement \( 0 = 6 \).

This equation is obviously false, as zero is never equal to six. A contradiction tells us that there is no set of numbers that satisfy both original equations. This contradiction happens because when we added the two equations, all variable terms canceled out, leaving behind an impossible statement.

Such contradictions indicate conflicting conditions in the original system, which usually represents parallel lines in a graph that never intersect. Thus, understand this concept helps students realize when a system of equations might not have a solution.

No Solution in Linear Algebra

In linear algebra, a system of equations may sometimes have no solution. This particular situation occurs when the equations represent parallel lines that never meet in a plane, as seen in this exercise. When we derive a contradiction (like \( 0 = 6 \)), it confirms that these lines do not intersect.

Having a system with no solution is known as an inconsistent system. No matter what values we substitute into the equations, the conditions cannot be satisfied simultaneously. In graphical terms, the gap between the parallel lines represents the lack of intersection points.

Recognizing systems with no solutions is important because it highlights boundaries and limitations within a set of conditions. Practical applications include understanding feasible regions in optimization problems and mechanical constraints in engineering design. It's key for students to grasp this idea to avoid assumptions of solutions where none exist.