Problem 45 - Precalculus: Mathematics for Calculus

Question

Solve the system, or show that it has no solution. If the system has infinitely many solutions, express them in the ordered-pair form given in Example 6. $$\left\\{\begin{array}{l} \frac{1}{2} x+\frac{3}{5} y=3 \\ \frac{5}{3} x+2 y=10 \end{array}\right.$$

Step-by-Step Solution

Answer

The system has infinitely many solutions expressed as \((x, \frac{30 - 5x}{6})\).

Step 1: Eliminate Fractions

To make calculations easier, let's first eliminate the fractions from both equations by finding a common denominator or multiplying through by the denominator. For the first equation: \(\frac{1}{2}x + \frac{3}{5}y = 3 \), multiply through by 10 to get \(5x + 6y = 30\). For the second equation: \(\frac{5}{3}x + 2y = 10\), multiply through by 3 to get \(5x + 6y = 30\).

Step 2: Check for Consistency

Now we have two equations: \(5x + 6y = 30\) and \(5x + 6y = 30\). Both equations are identical. This means every solution of the first equation is also a solution of the second.Since both equations are identical, the system is consistent and dependent, indicating there are infinitely many solutions.

Step 3: Find the General Solution

To express the solution in ordered-pair form, we solve for one variable in terms of the other. Let's solve for \(y\) in terms of \(x\). Using the first equation \(5x + 6y = 30\), we find:\[ 6y = 30 - 5x \]\[ y = \frac{30 - 5x}{6} \]Therefore, the solutions can be expressed in ordered-pair form as \((x, \frac{30 - 5x}{6})\), where \(x\) is any real number.

Key Concepts

Solutions of Linear Systems

Linear systems consist of two or more linear equations involving the same set of variables. The goal is to find values for these variables that satisfy all equations simultaneously. For example, in the system provided, you have two linear equations with variables \(x\) and \(y\). Most commonly, solutions fall into one of three categories:

• One solution: The system's equations intersect at a single point, giving one unique solution.
• No solution: The equations are parallel and never meet.
• Infinitely many solutions: The equations are dependent and represent the same line.

Each type represents the relationship between the equations' graphs. Solving a linear system involves various methods like substitution, elimination, or matrix operations, aimed at finding this relationship and identifying the solutions.

Infinitely Many Solutions

When a linear system has infinitely many solutions, it means the equations describe the same line. Thus, every point on this line is a solution. In the provided example, once the fractions were eliminated and the equations were simplified, both resulted in identical equations, \(5x + 6y = 30\). This implies that one equation is simply a scaled version of the other and doesn't introduce new information.
To describe the solutions, we express one variable in terms of the other, as shown in the step-by-step solution. Here, we solve for \(y\) and represent the solutions using ordered pairs like \((x, \frac{30 - 5x}{6})\), with \(x\) as a free variable. Hence, the system has an infinite number of solutions, with each point on the line being a solution set.

Dependent System

A dependent system occurs when two or more equations represent the same line. These equations provide the same information about the relationship between variables. In a geometric perspective, the graphs of these equations overlap completely as they are identical, which is why a dependent system always yields infinitely many solutions.
In our example, both simplified equations were \(5x + 6y = 30\), meaning any solution of one equation is a solution for the other. Recognizing a dependent system is crucial as it saves unnecessary further calculations for solutions. Instead of searching for distinct solutions, we note the relationship and express it in a generalized form, allowing for an understanding of how one variable depends on the other, thus providing clarity on the infinite nature of the solutions.