Problem 75 - Thinking Mathematically

Question

Make Sense? In Exercises 75-78, determine whether each statement makes sense or does not make sense, and explain your reasoning. Even if a linear system has a solution set involving fractions, such as \(\left\\{\left(\frac{8}{11}, \frac{43}{11}\right)\right\\}\), I can use graphs to determine if the solution set is reasonable.

Step-by-Step Solution

Answer

The statement does make sense because linear systems can be graphed in order to verify the reasonability of their solutions, regardless of whether they involve fractions or not.

Step 1: Understanding The Problem

The exercise is asking if a linear system, even with fractional solutions, can be verified for reasonability using a graph.

Step 2: Reasoning

Linear systems can be graphed on a coordinate plane irrespective of whether their solutions are fractional or not. Drawing a graph allows one to visually determine the point(s) of intersection of the linear equations in a system, which are essentially their solutions. So yes, even if the solution consists of fractions, it can still be plotted on a graph, and by visually inspecting the graph, the reasonability of the solution set can be verified.

Step 3: Conclusion

Based on the above reasoning, we can deduce that the statement does make sense. Even if the solution set of a linear system involves fractions, such as \(\left\{\left(\frac{8}{11}, \frac{43}{11}\right)\right\}\), it is indeed possible to use graphs to determine if the solution set is reasonable.

Key Concepts

Solution Set

In mathematics, a solution set refers to the collection of all possible solutions that satisfy a given equation or system of equations. For linear systems, the solution set can often be visualized as a point of intersection between two or more lines on a graph.
If you have a system of linear equations, each solution corresponds to a point where all the equations intersect.

• When there is one point of intersection, the solution set contains exactly one pair of values, usually written in the form \(x, y\).
• If the lines are parallel and never meet, there is no solution set.
• Should the lines coincide, so they are the same line, the solution set could be described as infinite, as every point on those lines represents a solution.

Different forms of solutions, such as fractions, are still part of the solution set and can be precisely plotted on a graph. This makes the solution set a powerful concept for checking the accuracy of proposed solutions.

Graphing Linear Equations

Graphing is a method used to represent equations by plotting points on a coordinate plane. Each linear equation can be rewritten in the slope-intercept form \(y = mx + b\) for easy graphing, where \(m\) is the slope and \(b\) is the y-intercept.
To graph a linear equation, follow these simple steps:

• Identify the slope and y-intercept from the equation.
• Start by plotting the y-intercept on the y-axis.
• Use the slope to find another point on the line. For a slope of \(m = \frac{a}{b}\), move "a" units up or down and "b" units left or right from the y-intercept.
• Draw a straight line through the points you have plotted. Extend it across the graph.

This graphical method provides a visual representation of potential solution sets. Even if solutions are fractions, the graph lets you compare and see if multiple lines intersect at well-defined points that match fractional coordinates.

Fractional Solutions

Fractional solutions in a linear system indicate that the solution coordinates are fractions instead of whole numbers. While fractions may seem complicated, they are a normal part of linear equations. Recognizing and understanding fractional solutions is essential as they add precision to mathematical solutions.
Let's look at why fractions occur:

• When equations in a system have coefficients or constants that don't resolve neatly into whole numbers, the intersection point of these equations is often a fractional value.
• For example, solving equations \(2x + 3y = 11\) and \(3x + 2y = 13\) can yield solutions like \(x = \frac{8}{11}\) and \(y = \frac{43}{11}\).

Fractions can be mapped on a graph just as easily as integers. This capability allows fractional solutions to be compared visually on a coordinate plane, providing a strong check on the validity of mathematical results.