Problem 25 - Precalculus: Mathematical 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 3. $$\left\\{\begin{array}{l} 6 x+4 y=12 \\ 9 x+6 y=18 \end{array}\right.$$

Step-by-Step Solution

Answer

The system has infinitely many solutions: \((x, \frac{6 - 3x}{2})\).

Step 1: Write the System of Equations

We have two equations in the system: 1. \(6x + 4y = 12\) 2. \(9x + 6y = 18\). We need to determine if there are any solutions, and whether there is one solution, no solution, or infinitely many solutions.

Step 2: Simplify the First Equation

Divide the first equation by 2 to simplify it: \(6x + 4y = 12\) becomes: \(3x + 2y = 6\).

Step 3: Simplify the Second Equation

Divide the second equation by 3 to simplify it: \(9x + 6y = 18\) becomes: \(3x + 2y = 6\).

Step 4: Compare the Simplified Equations

After simplification, both equations are identical: \(3x + 2y = 6\). This means that the two original equations are the same line and therefore have infinitely many solutions.

Step 5: Express the Solutions

We know \(3x + 2y = 6\) is the line representing the solutions. Let's express \(y\) in terms of \(x\): \(2y = 6 - 3x \) \(y = \frac{6 - 3x}{2}\) Therefore, the solutions in ordered-pair form are: \((x, \frac{6 - 3x}{2})\).

Key Concepts

Infinitely Many Solutions

In the realm of systems of equations, one possible outcome is having infinitely many solutions. When two equations represent the same line, they overlap entirely. This happens because every point on the line satisfies both equations simultaneously.

In our problem, once the equations were simplified, they both reduced to \(3x + 2y = 6\). This implies they describe the same line. Therefore, every point that satisfies the equation \(3x + 2y = 6\) is a solution to the system.

To express this solution set in ordered-pair form, as shown in the example, we solve for one variable in terms of the other. This provides a way to describe all points — for instance, solve for \(y\) to get \(y = \frac{6 - 3x}{2}\). Now, every point \((x, \frac{6 - 3x}{2})\) along the line is a solution, hence infinitely many solutions.

Linear Equations

Linear equations are the foundation of algebra. They are equations involving variables raised only to the first power. Their graphs are straight lines, which can be described with an equation in the form \(ax + by = c\).

In a system of linear equations, each equation represents a line. When two lines are part of the same system, they can intersect, be parallel, or coincide completely. The unique intersection of lines can give zero, one, or infinitely many solutions.

For the given exercise, the equations \(6x + 4y = 12\) and \(9x + 6y = 18\) represent linear equations. Simplifying these linear equations leads to \(3x + 2y = 6\) for both, indicating they represent the same line and hence have infinitely many solutions.

Solving Systems of Equations

Solving systems of equations involves finding values for variables that satisfy all equations simultaneously. There are several methods to solve them, such as substitution, elimination, or graphing, each offering a different approach.

In this problem, however, both the substitution and elimination methods simplify into recognizing the same line. When simplified, it became clear that both equations led to \(3x + 2y = 6\). This makes solving straightforward without further calculations since both lines overlapped.

Since the system of equations shares this identical line, it means you have infinitely many solutions. This is a special scenario of dependent systems, where an infinite number of \((x, y)\) pairs can satisfy the equation \(3x + 2y = 6\). To express this set of solutions, represent one variable in terms of the other, as seen in the solution.