Problem 15 - Precalculus: Functions and Graphs

Question

Solve the system. $$\left\\{\begin{aligned} 3 m-4 n &=2 \\ -6 m+8 n &=-4 \end{aligned}\right.$$

Step-by-Step Solution

Answer

The system has infinitely many solutions because the two equations are identical.

Step 1: Identify the Equations

We have two equations: 1) \( 3m - 4n = 2 \) 2) \( -6m + 8n = -4 \).

Step 2: Simplify the Second Equation

Observe that the second equation is \(-6m + 8n = -4\). This equation can be simplified by dividing every term by \(-2\):\[-6m + 8n = -4\]\[3m - 4n = 2\]This results in the same equation as the first, \(3m - 4n = 2\).

Step 3: Analyze the System

Since both equations are essentially the same, they represent the same line. Thus, the system has infinitely many solutions, as any point along the line \(3m - 4n = 2\) will satisfy both equations.

Key Concepts

Linear Equations

A linear equation is an equation that forms a straight line when graphed. In a two-variable context, like the one seen in this system, it represents a relationship between two variables, often denoted as \( m \) and \( n \). A linear equation can generally be expressed in the form \( ax + by = c \), where \( a \), \( b \), and \( c \) are constants. These equations are fundamental because they model many real-world situations through a constant rate of change between the variables. In the given exercise, the equations \( 3m - 4n = 2 \) and \( -6m + 8n = -4 \) are both linear and represent potential relationships between \( m \) and \( n \).Understanding linear equations involves recognizing that each equation represents a line. The solution to a system of linear equations is any point where these lines intersect, if they do. If there is an intersection at one point, there is one solution. If the lines are parallel and distinct, there are no solutions. And if the lines are identical, there are infinite solutions.

Infinite Solutions

Infinite solutions occur when two lines (equations) overlap completely, meaning every point on one line is also on the other. When this happens, the system of equations is said to be dependent.In the case of the given exercise, after simplifying the second equation, we find it duplicates the first: both simplify to \( 3m - 4n = 2 \). This indicates that instead of two distinct lines, we have one line represented twice.

• This phenomenon tells us that every point on the line meets the conditions set by both equations.
• Graphically, it would appear as one line seemingly drawn on top of itself.
• Infinite solutions mean the system is not fully independent, providing overlapping or the same constraints twice.

You can think of infinite solutions as a linear line that goes forever in both directions, encompassing all infinite values of \( m \) and \( n \) that satisfy the line's equation.

Simplification of Equations

Simplifying equations is a key step in solving systems because it can often reveal hidden characteristics of the system, as seen in our exercise.Simplification entails breaking down the equations into their simplest form, generally by finding a common factor or using arithmetic operations to combine or reduce terms. In the exercise, we simplified the second equation by dividing each term by \(-2\).

• This process resulted in the simplified equation \( 3m - 4n = 2 \), demonstrating its equivalence to the first equation.
• Simplification exposed the identity of the equations, either revealing that they describe the same line or are better understood in a different form.
• Effective simplification can prevent mistaken interpretations by revealing the essential nature and relationships in the system.

Grasping the power of simplifying equations is crucial, enabling easier resolution of systems and a deeper understanding of their underlying dynamics.