Problem 83 - College Algebra with Modeling and Visualization

Question

Solve the system, if possible. $$ \begin{array}{rr} 7 x-3 y= & -17 \\ -21 x+9 y= & 51 \end{array} $$

Step-by-Step Solution

Answer

The system has infinitely many solutions.

Step 1: Identify System Properties

The given system of linear equations is: \( 7x - 3y = -17 \) and \( -21x + 9y = 51 \). Observe that the second equation is a multiple of the first equation. Specifically, the second equation is obtained by multiplying the first equation by -3, indicating that the two equations are dependent.

Step 2: Simplify the Second Equation

Multiply the first equation by -3 to compare it exactly with the second equation. Accordingly, we get \( -21x + 9y = 51 \), which is identical to the second equation, confirming that these equations are essentially the same or scalar multiples of each other.

Step 3: Analyze the System

Since both equations describe the same line, there are infinitely many solutions, as every point on the line is a solution. The system is dependent, leading to infinitely many solutions along this line.

Key Concepts

Dependent Systems

A dependent system in linear equations occurs when the equations are multiples of each other. This means the system doesn't represent two distinct lines, but rather the same line presented in different forms. For example, in the given system:\[7x - 3y = -17 \text{ and } -21x + 9y = 51\]The second equation is simply the first equation multiplied by \(-3\). This indicates that both sets of equations describe the same line.

In a dependent system:

• The equations provide no new information from one another.
• They are consistent, meaning they have solutions that satisfy both equations.
• They cannot be graphically represented as two lines intersecting or running parallel. Rather, they overlap perfectly.

Dependent systems are a crucial concept for finding when two equations essentially describe one line, not two different lines intersecting or being parallel.

Infinite Solutions

When dealing with a system of equations that is dependent, we encounter infinite solutions. This happens because the equations describe the same line graphically. Since every point on this line is a solution, there isn't just one point of intersection, but a whole line of solutions. Let's look at the properties and implications:

• Any value for one variable can be matched to a corresponding value of the other variable to satisfy both equations.
• This line of solutions can be expressed in parametric form, where one variable is written in terms of another.
• Graphically, it means that, instead of two lines crossing at a single point, one line is repeated.

This concept is important when determining the nature of solutions to systems of linear equations and understanding how equations relate to each other.

Linear Combinations

Linear combinations help to determine if systems such as the one given are dependent. It involves checking if one equation can be "built" by multiplying the other by a constant. In the presented system, we clearly observe this:

• The second equation, \(-21x + 9y = 51\), is obtained by multiplying the entire first equation, \(7x - 3y = -17\), by \(-3\).
• It shows that the same relationship between \(x\) and \(y\) holds in both equations, just presented in different magnitudes.
• Such an approach ensures we effectively identify dependent relationships between equations, diagnosing if equations are essentially the same.

This concept helps students understand deeper mathematical relationships between equations and prepares them for complex problem-solving involving linear equations.