Problem 20 - Contemporary Precalculus

Question

Use the elimination method to solve the system. $$\begin{aligned}&2 x-3 y=7\\\&6 x-9 y=1\end{aligned}$$

Step-by-Step Solution

Answer

$$\begin{aligned}&2 x - 3 y = 7\\&6 x - 9 y = 1\end{aligned}$$ Answer: The system of linear equations has no solution using the elimination method.

Step 1: Write the given system of equations

Write the system of linear equations as provided in the exercise: $$\begin{aligned}&2 x - 3 y = 7\\&6 x - 9 y = 1\end{aligned}$$

Step 2: Multiply one or both equations to make the coefficients of one variable equal

In order to eliminate one of the variables, we need to make the coefficients of either x or y equal. In this case, we will multiply the first equation by 3 to match the coefficient of x in the second equation: $$\begin{aligned}&(3)(2 x - 3 y) = (3)(7)\\&6 x - 9 y = 1\end{aligned}$$ Which yields: $$\begin{aligned}&6 x - 9 y = 21\\&6 x - 9 y = 1\end{aligned}$$

Step 3: Eliminate one variable

To eliminate x, subtract the second equation from the first equation: $$\begin{aligned}(6 x - 9 y) - (6 x - 9 y) = 21 - 1\end{aligned}$$ This results in: $$0 = 20$$

Step 4: Check for consistency

Since we have obtained a false statement (0 ≠ 20) in our elimination process, the given system of equations is inconsistent, meaning it has no solution.

Step 5: Write the final answer

The given system of linear equations has no solution, so the final answer is no solution.

Key Concepts

System of Linear Equations

A system of linear equations consists of two or more linear equations that have common solutions. These equations represent lines in a two-dimensional space or planes in higher dimensions. Solving such a system means finding all the values of the variables that satisfy all the equations simultaneously.

For instance, the exercise presents a system with two equations involving two variables, x and y, and the goal is to find the values of x and y that satisfy both equations. Systems like these can have one unique solution, infinitely many solutions, or no solution at all, and identifying which case applies is crucial to correctly solving the system.

Inconsistent System

An inconsistent system of equations is one in which no single set of values satisfies all the equations at the same time. This usually happens when the lines represented by the equations have different slopes and do not intersect, or when they are parallel.

In the exercise provided, after following the elimination method, we obtain an equation that is false (0 = 20). This indicates that our original system of equations is inconsistent, as the lines would not intersect at any point, and therefore, there is no solution for the system.

Recognizing Inconsistent Systems

It's important to recognize the signs of inconsistency; one clear sign is arriving at a false or contradictory statement when attempting to solve the equations.

Solving Linear Systems

Solving linear systems involves finding the variable values that satisfy all given equations. There are several methods to solve these systems, such as graphing, substitution, and elimination.

The elimination method, as used in the textbook exercise, involves manipulating the equations to eliminate one variable, making it easier to find the value(s) of the remaining variable(s). This method requires careful arithmetic and sometimes involves multiplying equations to align the coefficients of variables to simplify the system.

Steps for Elimination Method

• Multiply or divide equations to obtain equal coefficients for one variable.
• Add or subtract equations to eliminate one variable.
• Solve for the remaining variable.
• Back-substitute to find the other variable's value if the system is consistent.
• Assess the results to determine if the system is consistent, inconsistent, or dependent.

When we come across an inconsistency or a dependent system (infinitely many solutions), the elimination method helps us conclude effectively.