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Chapter 7: Problem 76

Make Sense? Determine whether each statement makes sense or does not make sense, and explain your reasoning. Each equation in a system of linear equations has infinitely many ordered-pair solutions.

Short Answer

Expert verified
The statement does not make sense. Even though every line in a Cartesian plane (thus every linear equation) has infinitely many ordered-pair solutions, a system of linear equations does not necessarily share the same solutions. The system itself may have a unique solution, infinite solutions (if equations are the same), or no solution at all.

Step by step solution

01

Understanding the Statement

A system of linear equations represents a set of straight lines on a Cartesian plane. Every point on each line is an ordered pair solution of that particular linear equation. It means any line in a plane has infinitely many points, hence, each linear equation has infinitely many ordered-pair solutions.
02

Understanding System of Linear Equations

When we have a 'system of linear equations', we are looking for the common points of intersection of all the lines in the system. This implies that we are trying to find the ordered-pair which satisfies all the equations in the system, not only one of them. So, in a system of linear equations, while each equation individually has infinite solutions, the system of equations may not.
03

Final Reasoning

In a system of linear equations, we are interested in the shared solutions of all equations, i.e., the intersection points. While it's true that each equation separately has an infinite number of solutions, the system as a whole does not. It might have a single solution (one intersection point), an infinite number of solutions (the equations are equivalent and represent the same line), or no solution (parallel lines with no intersection). So the statement doesn't make sense when related to the system of linear equations as a whole.

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Most popular questions from this chapter

The graphs of solution sets of systems of inequalities involve finding the intersection of the solution sets of two or more inequalities. By contrast, in Exercises 43-44, you will be graphing the union of the solution sets of two inequalities. Graph the union of \(y>\frac{3}{2} x-2\) and \(y<4\).

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