Problem 19 - 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. $$\left\\{\begin{array}{r} 3 x-2 y=8 \\ -6 x+4 y=16 \end{array}\right.$$

Step-by-Step Solution

Answer

The system has infinitely many solutions: \((x, \frac{3}{2}x - 4)\).

Step 1: Analyze the System of Equations

The given system of equations is \( \begin{cases} 3x - 2y = 8 \ -6x + 4y = 16 \end{cases} \). We need to determine whether it has a unique solution, infinitely many solutions, or no solution.

Step 2: Simplify the Second Equation

We can multiply the first equation by 2 to compare it with the second equation: \( 2(3x - 2y) = 2(8) \), resulting in \( 6x - 4y = 16 \).

Step 3: Compare the Equations

Notice that the second equation \( -6x + 4y = 16 \) is the negative of the simplified version of the first equation, as \( 6x - 4y = 16 \). This indicates the two equations are equivalent.

Step 4: Determine the Relationship Between the Equations

Since one equation is a multiple of the other, the two equations are dependent, indicating infinitely many solutions.

Step 5: Express the Solution in Ordered-Pair Form

To express the solutions in ordered-pair form, take one equation (e.g., \( 3x - 2y = 8 \)) and solve for one variable. Solving for \( y \) gives us: \( y = \frac{3}{2}x - 4 \). Therefore, the ordered-pair form is \( (x, \frac{3}{2}x - 4) \).

Key Concepts

Infinite Solutions

When dealing with systems of linear equations, it's crucial to understand what is meant by having **infinite solutions**. This occurs when any values for the variables involved satisfy the equations in the system.

In our example, we initially have two equations:

• \( 3x - 2y = 8 \)
• \( -6x + 4y = 16 \)

After simplifying, both equations represent the same line. When graphed, they lie on top of each other. This translates to infinite solutions because every point on the line is a solution.Explaining intuitively:

• Every **x-value** on the line has a corresponding **y-value** that makes both original equations true.
• Because they are the same line, there are countless points (x, y) that will work.

Understanding this concept helps in recognizing when a system of equations has more than one solution.

Dependent Equations

**Dependent equations** are equations that are algebraically related, meaning that one equation is just a different expression of the other. This indicates that the system of equations represents the same line when graphed.

In our case, by multiplying the first equation by 2, we achieved an equation that is the negative of the second:

• Original: \( 3x - 2y = 8 \)
• Adjusted: \( 6x - 4y = 16 \)

Both equations reflect the same geometric line, which confirms they are dependent. Here are some more key points:

• Dependent systems always indicate a relationship between the equations that result in them being equivalent.
• This means they share all solutions, leading directly to them having infinite solutions.

Identifying dependency is crucial because it simplifies recognizing redundant equations in the system.

Ordered-Pair Form

In cases of infinite solutions, it’s often useful to express them in what’s called **ordered-pair form**. This gives the solutions as a set of points (x, y) that satisfies the system of equations.

Using our example, once we clarified that the system has infinite solutions, we can find an ordered-pair by isolating one variable in terms of another:

• From \( 3x - 2y = 8 \), solving for \( y \) gives us: \( y = \frac{3}{2}x - 4 \).

Thus, the ordered-pair form of these solutions is \((x, \frac{3}{2}x - 4)\):

• Each **x** value corresponds to a unique **y** value on the line.
• This form is particularly useful because it allows you to quickly generate points on the graph.

This method is a straightforward way to visually and algebraically display infinite solutions on a graph.