Problem 19 - Algebra and Trigonometry

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. $$\left\\{\begin{aligned} 3 x-2 y &=8 \\\\-6 x+4 y &=16 \end{aligned}\right.$$

Step-by-Step Solution

Answer

The system has no solution; the lines are parallel.

Step 1: Observe and Analyze the System

Examine the system of equations:1. \(3x - 2y = 8\) 2. \(-6x + 4y = 16\).Notice that the second equation is a multiple of the first equation.

Step 2: Simplify the Second Equation

Simplify the second equation by dividing every term by \(-2\):\(-6x + 4y = 16\) becomes \(3x - 2y = -8\).

Step 3: Compare the Equations

Now, compare the simplified system:1. \(3x - 2y = 8\) 2. \(3x - 2y = -8\).This reveals that both lines are parallel and distinct.

Step 4: Conclude No Solution Exists

Since the two equations represent parallel lines with different y-intercepts, they never intersect. Hence, there is no solution to the system.

Key Concepts

Linear Equations

Linear equations are mathematical expressions involving constants and variables, typically in the form of ax + by = c. Here, a and b are coefficients, x and y are variables, and c is a constant. These equations are called "linear" because they graph as straight lines on a coordinate plane.
An example of a linear equation includes 3x - 2y = 8. Notice how it expresses a relationship between x and y.

• When solving linear equations, we aim to find all possible pairs of x and y that satisfy both equations simultaneously.
• This involves rearranging, simplifying, or using various algebraic methods to find points of intersection on the graph.

Linear equations often form part of a system of equations, where solving them requires finding common solutions that satisfy each equation in the system.

Parallel Lines

Parallel lines are lines in the same plane that never intersect. They remain a constant distance from one another and have the same slope but different y-intercepts. In graphing linear equations, parallel lines indicate that the system has no points that satisfy both equations simultaneously.
When examining two linear equations, such as:

• 3x - 2y = 8
• 3x - 2y = -8

We note that both have identical slope and differing y-intercepts; hence, they're parallel. This aligns with our observation that:

• The slope here is given by the ratio of coefficients from x and y terms, ensuring lines do not meet.
• Differing constants (8 and -8) confirm they are distinct and parallel.

Thus, understanding parallelism in linear equations helps predict the existence of solutions.

No Solution

A system of linear equations with no solution is one where the equations represent parallel, non-intersecting lines. This situation means no point is shared between the equations, hence no solution exists for the system.
In our example, we've seen how:

• The system 3x - 2y = 8 and 3x - 2y = -8 shows two equations that are parallel.
• Since they have different y-intercepts but equal slopes, they will never intersect.

This scenario highlights why the system has no solution, as parallel lines do not converge at any point in the plane. Recognizing no solution involves checking the slope and intercept characteristics of each line in the system.

Algebraic Methods

Algebraic methods are essential for solving systems of equations. These methods allow us to manipulate and simplify equations to determine if solutions exist. Common algebraic approaches include substitution, elimination, and comparison.
In our exercise, we applied the comparison method:

• Initially, we observed that the given system included 3x - 2y = 8 and -6x + 4y = 16.
• We simplified the second equation by dividing all terms by -2, leading to 3x - 2y = -8.

Using simplification, we determined the equations were parallel. Simple algebraic adjustments in equations reveal many characteristics without graphical methods. Practicing these methods sharpens skills in identifying and solving linear systems and understanding solutions' existence or absence.