Problem 27 - 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) $$ \left(\begin{array}{r} 5 x-2 y=1 \\ 10 x-4 y=7 \end{array}\right) $$

Step-by-Step Solution

Answer

No solution; the system is inconsistent (parallel lines).

Step 1: Multiply Equations for Matching Coefficients

Start by examining the coefficients of both equations. The second equation is a multiple of the first equation.**Equation 1**: \(5x - 2y = 1\)**Equation 2**: \(10x - 4y = 7\)Notice that \(10x - 4y\) is exactly double of \(5x - 2y\), so multiply the first equation by 2:\[2(5x - 2y) = 2(1)\]Which simplifies to:\(10x - 4y = 2\)

Step 2: Subtract Equations to Eliminate Variables

Now that both equations have identical terms for \(x\) and \(y\), subtract the modified **Equation 1** from **Equation 2**:**Equation 2**: \(10x - 4y = 7\)Modified **Equation 1**: \(10x - 4y = 2\)Subtract:\[(10x - 4y) - (10x - 4y) = 7 - 2\]This results in:\(0 = 5\)

Step 3: Interpret the Result for Consistency

A result of \(0 = 5\) is a contradiction, indicating that there is no solution to the system. The two equations are parallel lines, which never intersect. This means the system of equations is inconsistent.

Key Concepts

Substitution method

The substitution method is a systematic approach to solving systems of linear equations. It involves solving one of the equations for one variable and substituting that expression into the other equation. This method is particularly useful when one of the equations can be easily solved for one variable. In such cases, substitution can simplify the system, allowing us to solve it more straightforwardly.

To use the substitution method, follow these general steps:

• Solve one of the equations for one variable in terms of the other.
• Substitute the expression obtained into the other equation.
• Solve the resulting single-variable equation.
• Substitute back to find the value of the other variable.

Although the substitution method isn't applied in this specific exercise's solution, understanding it is key. It can be a helpful tool when the system has a straightforward equation that isolates one variable easily.

Elimination method

The elimination method is often the go-to strategy when dealing with systems of equations, especially when coefficients align neatly or can be adjusted to align. This method works by adding or subtracting equations to eliminate one of the variables, simplifying the system to a single equation and one variable.

Here's how the elimination method came to play in this exercise:

• The coefficients of the first equation were adjusted to match the second equation. This was achieved by multiplying the entire equation by a number that made both equations equivalent in terms of one set of variables.
• Once the equations were aligned, they were subtracted to eliminate both variables.
• This subtraction resulted in a notably odd outcome due to the complete cancellation of variables, leading us to the contradiction identified in the step-by-step solution.

The elimination method effectively simplified the process, allowing straightforward detection of inconsistency in the given system.

Inconsistent systems

Understanding inconsistent systems is crucial in the study of linear equations. An inconsistent system occurs when the equations represent parallel lines, meaning they never intersect. As a result, there are no solutions that satisfy both equations simultaneously, reflecting a logical contradiction.

In the exercise above, the discovery of an inconsistent system was through the elimination process, where the subtraction of aligned equations led to an impossible statement, such as \(0 = 5\). This kind of result unmistakably indicates parallel lines in the graphical representation of the system.

• In geometric terms, the lines have the same slope but different y-intercepts.
• An inconsistent system typically results when one equation is a scalar multiple of another, yet with a differing constant on the right-hand side.
• Recognizing these signs helps to quickly identify systems where no solution exists, simplifying the analysis of the system.

Being familiar with inconsistent systems ensures you interpret the results accurately, whether in algebraic operations or graphical assessments.