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Chapter 2: Problem 28

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. A system composed of two linear equations must have at least one solution if the straight lines represented by these equations are nonparallel.

Short Answer

Expert verified
The statement is true. A system composed of two linear equations must have at least one solution if the straight lines represented by these equations are nonparallel. This is because nonparallel lines can either intersect at a single point (one unique solution) or be coincident and overlap completely (infinite solutions). In both cases, there is at least one point that satisfies the equations in the system.

Step by step solution

01

Understand the types of solutions for a system of two linear equations

A system of two linear equations can have three possible types of solutions: 1. One unique solution: This occurs when the two lines intersect at a single point. This is the point that satisfies both equations. 2. No solutions (Inconsistent): This occurs when the two lines are parallel and do not intersect. In this case, there is no point that satisfies both equations. 3. Infinite solutions (Dependent): This occurs when the two lines are coincident, which means they are the same line. Every point on the line satisfies both equations.
02

Relate the given condition to the types of solutions

According to the given statement, the two lines are nonparallel. This means that the only possible types of solutions are either one unique solution or infinite solutions. Therefore, in this case, the system must have at least one solution.
03

Conclusion

The statement is true. A system composed of two linear equations must have at least one solution if the straight lines represented by these equations are nonparallel. This is because nonparallel lines can either intersect at a single point (one unique solution) or be coincident and overlap completely (infinite solutions). In both cases, there is at least one point that satisfies the equations in the system.

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