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Chapter 5: Problem 52

Determine the Number of Solutions of a Linear System Without graphing the following systems of equations, determine the number of solutions and then classify the system of equations. $$ \left\\{\begin{array}{l} y=\frac{1}{3} x+2 \\ x-3 y=9 \end{array}\right. $$

Short Answer

Expert verified
The system has no solutions and is inconsistent.

Step by step solution

01

Write Down the Equations

The given system of equations is:1. \( y = \frac{1}{3} x + 2 \)2. \( x - 3y = 9 \)
02

Substitute Equation 1 into Equation 2

Substitute \( y = \frac{1}{3} x + 2 \) into the second equation:\( x - 3\left( \frac{1}{3} x + 2 \right) = 9 \)
03

Simplify the Equation

Distribute the \( -3 \):\( x - x - 6 = 9 \). This simplifies to: \(-6 = 9 \).
04

Analyze the Result

The equation \(-6 = 9 \) is a contradiction, which means it is never true.
05

Classify the System

Since the result is a contradiction, the system of equations has no solutions. It is therefore inconsistent.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

linear equations
Linear equations are equations of the first degree, meaning they have variables raised only to the power of one. Each equation typically forms a straight line when graphed on a coordinate plane. A system of linear equations consists of two or more linear equations using the same set of variables. Here are some key points to remember about linear equations:

They often appear in the form \( ax + by=c \)
The graph of linear equations is a straight line
Solutions to linear equations are combinations of variables that satisfy all the equations in the system simultaneously.
inconsistent system
An inconsistent system of equations means that there are no solutions that satisfy all equations simultaneously. This occurs when the equations represent parallel lines that never intersect. No single point exists where all the equations hold true. In other words, you cannot find a single set of values for the variables that make all the equations true at the same time. Identifying an inconsistent system involves:

Recognizing contradictory results that can't be true, like \(-6 = 9 \)
Finding parallel lines that do not intersect.
algebraic substitution
Algebraic substitution is a method used to solve systems of equations by solving one equation for one variable and then substituting that expression into the other equation. This method helps break down complex systems into simpler, more manageable steps. For the given exercise:

We start with the system: \( y = \frac{1}{3} x + 2 \) and \( x - 3y = 9 \).
Solve the first equation for \( y \): \( y = \frac{1}{3} x + 2 \).
Substitute the expression for \( y \) into the second equation.

This substitution provides us a simpler equation to solve, and ultimately check for contradictions.
contradiction in equations
A contradiction in equations arises when simplifying an equation leads to an impossible statement, such as \(-6 = 9 \). This indicates the system is inconsistent. In the exercise, substituting \( y = \frac{1}{3} x + 2 \) into \( x - 3y = 9 \) gave us: \( x - x - 6 = 9 \). Simplifying this results in: \(-6 = 9 \), which is a contradiction. This contradiction tells us that no set of \(x, y \) values can satisfy both equations at the same time, confirming the inconsistency of the system. Key points include:

Looking out for simplified equations resulting in false statements.
Recognizing that contradictions mean no solutions exist for the system.

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