Problem 95 - Algebra and Trigonometry Real Mathematics, Real People

Question

Think About It Find all value(s) of \(k\) for which the system of linear equations \(\left\\{\begin{array}{c}x+3 y=9 \\ 2 x+6 y=k\end{array}\right.\) has (a) infinitely many solutions and (b) no solution.

Step-by-Step Solution

Answer

The system has infinitely many solutions for \(k = 18\) and no solution for \(k \neq 18\).

Step 1: Manipulate the first equation

We can manipulate the first equation to fit the form of the second equation. Multiply the first equation by 2: \(2x + 6y = 18\).

Step 2: Case (a): Infinitely many solutions

Now, if we equate this to the second equation, we can see that if \(k = 18\), both equations are identical. This means, for \(k = 18\), the system has infinitely many solutions.

Step 3: Case (b): No solution

For the system to have no solution, the lines drawn by the equations must be parallel and not intersect. The lines are already parallel. For them not to intersect, the constant term must be different. If \(k\) is any number other than 18, the lines do not intersect, therefore giving no solution.

Key Concepts

Infinitely Many Solutions

When a system of linear equations has infinitely many solutions, it means all the equations describe the same line. Hence, every point on that line satisfies all the equations in the system. In simpler terms, if you were to graph the equations, they would lie atop one another, appearing as a single line.

For this to happen, the equations must be identical, except that one might be a multiple of the other. In our example, the system given is:

• Equation 1: \(x + 3y = 9\)
• Equation 2: \(2x + 6y = k\)

The first equation, when multiplied by 2, becomes \(2x + 6y = 18\). Thus, it is evident that if \(k = 18\), both equations become the same, indicating infinitely many solutions.

Parallel Lines

Parallel lines in a coordinate system are lines that never meet. They have the same slope but different y-intercepts.

In a system of linear equations, if the equations represent parallel lines, it means they will never intersect. This characteristic is crucial for understanding when two equations might result in no solutions, also known as having an inconsistent system.

• The slope of a line from the equation \(ax + by = c\) is \(-\frac{a}{b}\).
• If two lines have the same slope but different intercepts, their graphically represented lines are parallel.

In our exercise, the lines from both equations derived have the same slope value of \(-\frac{1}{3}\), thus they are parallel.

Inconsistent System

An inconsistent system of equations is one that has no solutions. This occurs when the equations describe parallel lines that never intersect.

If the symbolization in the equations is such that their slopes are equal but intercepts differ, there will be no solutions. For our system,

• The first equation, after manipulation, can be written as \(2x + 6y = 18\).
• The second equation is \(2x + 6y = k\).

To have no solution, \(k\) just needs to differ from 18. Any other value for \(k\) implies the lines stay parallel and distinct, thereby ensuring no intersection, which characterizes the system as inconsistent.