Problem 28 - Intermediate Algebra

Question

For Problems 19-48, solve each system by using either the substitution or the elimination-by-addition method, whichever seems more appropriate. (Objective 2) $$ \left(\begin{array}{c} y=\frac{2}{3} x-4 \\ 2 x-3 y=1 \end{array}\right) $$

Step-by-Step Solution

Answer

The system has no solution; it is inconsistent.

Step 1: Substitution Method Setup

We are given the system of equations: \( y = \frac{2}{3}x - 4 \) and \( 2x - 3y = 1 \). Since the first equation expresses \( y \) in terms of \( x \), we can use the substitution method. Substitute \( y = \frac{2}{3}x - 4 \) into the second equation.

Step 2: Substitute and Simplify

Substitute \( y \) into the second equation: \[2x - 3(\frac{2}{3}x - 4) = 1\]Distribute the \( -3 \) inside the parentheses:\[2x - 2x + 12 = 1\]Notice that \( 2x - 2x = 0 \), so we are left with:\[12 = 1\]

Step 3: Analyze the Result

Since the equation \( 12 = 1 \) is a contradiction and not possible, this indicates that there are no values of \( x \) and \( y \) that can satisfy both equations simultaneously. Therefore, the system is inconsistent.

Key Concepts

Substitution Method

The substitution method is a technique used to solve systems of equations. It involves replacing one variable with an equivalent expression from another equation. This method is particularly useful when one of the equations in the system already solves for one of the variables, making it easy to substitute into the other equation.

Here's a step-by-step guide on how it works:

• Identify one of the equations that is easier to solve for one variable, such as a simple equation for \(y\) or \(x\).
• Substitute this expression into the other equation in place of the variable. This reduces the system to a single equation with one variable.
• Solve the resulting equation for the unknown variable.
• Back-substitute the found value into one of the original equations to solve for the other variable.

In our example, the first equation \(y = \frac{2}{3}x - 4\) is substituted into the second equation \(2x - 3y = 1\). However, keep an eye out for contradictions which may lead you to different conclusions about the nature of the system.

Elimination Method

The elimination method, also known as the addition method, is another common technique for solving systems of linear equations. This method involves combining equations in the system to eliminate one of the variables, allowing you to solve for the remaining variable. It's particularly useful when the coefficients of one of the variables are easily manipulated to become equal between the two equations.

To apply the elimination method, follow these steps:

• Multiply each equation by suitable numbers to align the coefficients of one of the variables.
• Add or subtract the equations from each other to eliminate the chosen variable.
• Solve the resulting equation for the remaining variable.
• Use the found value to solve for the eliminated variable in one of the original equations.

This method often provides an efficient way to find solutions when substitution is cumbersome or when both equations are of similar complexity. In cases where you encounter a contradiction or an identical outcome, the system may be inconsistent or have infinitely many solutions, indicating a need for further analysis.

Inconsistent Systems

An inconsistent system is a system of equations that has no solutions. This occurs when the equations represent parallel lines, which never intersect. In mathematical terms, these lines have the same slope but different y-intercepts, leading to a scenario where there are no common solutions for the variables.

In the step-by-step solution provided in this problem, the inconsistency is identified when a contradiction arises, specifically when solving the system yields a false statement like \(12 = 1\). Such contradictions confirm that there are no solutions to the system of equations.

Understanding whether a system is inconsistent includes:

• Equations simplifying to a contradiction during substitution or elimination.
• Observation of identical slopes but different y-intercepts in linear equations.
• Verification by graphically representing the equations.

Recognizing an inconsistent system is an essential part of solving systems of equations, ensuring that efforts are focused appropriately and solutions are correctly interpreted.