Problem 18 - Algebra and Trigonometry with Analytic Geometry

Question

Solve the system. $$ \left\\{\begin{array}{r} x-5 y=2 \\ 3 x-15 y=6 \end{array}\right. $$

Step-by-Step Solution

Answer

The system has infinitely many solutions.

Step 1: Identify the Type of System

The given system consists of two linear equations. We have:1. Equation 1: \(x - 5y = 2\)2. Equation 2: \(3x - 15y = 6\).

Step 2: Check Equations for Proportionality

Multiply Equation 1 by 3 to compare it with Equation 2:\[3(x - 5y) = 3(2)\]This simplifies to:\(3x - 15y = 6\), which is exactly the same as Equation 2. Both equations represent the same line.

Step 3: Determine the Nature of the System

Since Equation 2 is a multiple of Equation 1, the system has infinitely many solutions. The two lines are coincident, meaning they lie on top of one another.

Key Concepts

Linear Equations

Linear equations are fundamental in mathematics and are equations where each term is either constant or the product of a constant with a single variable. They take the form \(ax + by = c\), where \(a\), \(b\), and \(c\) are constants. In a system of linear equations, we are looking at multiple such equations together. Each equation represents a line when graphed on a coordinate plane.
Understanding linear equations helps us solve systems, as each equation gives us a unique line, and the solution to the system tells us where these lines intersect (or meet). In some cases, such as this exercise, the lines may even overlap completely, representing the same line.

• Solution: Find where lines intersect.
• Graphically: Pick points and draw lines based on equations.
• Algebraic Skills: Use methods like substitution or elimination for solutions.

Infinitely Many Solutions

A system of equations has infinitely many solutions when not just one or a few points satisfy all the equations, but an entire set of points. This occurs when both equations represent the same line.
In mathematical terms, if one equation is a multiple of the other, they overlap completely. As a result, every point on the line is a solution to the system.

• Mathematical Observation: Equations are multiples.
• Geometric Interpretation: Lines coincide.
• Practical Implication: No unique solution unless there's more information.

Recognizing infinitely many solutions can prevent unnecessary calculations. It signals that the problem is more about understanding the equations’ relationship than solving for a unique answer.

Coincident Lines

Coincident lines occur when two lines share every point. This means they exactly lie on top of each other when drawn on a graph. This is the strongest form of overlap in linear equations.
In practical terms, when you solve a series of equations and find them to be coincident, you've essentially found the same line expressed differently. Each form, multiplication, or simplification of one equation leads back to the same line.

• Graphically: Both lines are indistinguishable.
• Equation Check: One is a multiple of the other.
• System Nature: Indicates infinite solutions with context.

Comprehending coincident lines helps in visualizing complex systems. It's a useful concept in understanding how different algebraic expressions can, in fact, represent the same geometric object.