Problem 52 - College Algebra with Modeling and Visualization

Question

If possible, solve the system of linear equations and check your answer. $$ \begin{aligned} 0.6 x-0.2 y &=2 \\ -1.2 x+0.4 y &=3 \end{aligned} $$

Step-by-Step Solution

Answer

The system is inconsistent and has no solution.

Step 1: Set Up the System of Equations

The given system of equations is: \[ \begin{aligned} 0.6x - 0.2y &= 2 \-1.2x + 0.4y &= 3 \end{aligned} \]Observe that the coefficients of the equations are decimals.

Step 2: Eliminate Decimals by Multiplying

To eliminate decimals, multiply the first equation by 10 and the second equation by 10 to obtain integers: \[ \begin{aligned} 6x - 2y &= 20 \-12x + 4y &= 30 \end{aligned} \]

Step 3: Multiply the First Equation to Align Coefficients

Multiply the entire first equation by 2 to facilitate elimination of the terms involving \(y\): \[ 12x - 4y = 40 \]

Step 4: Add the Equations to Eliminate 'y'

Add the modified first equation \(12x - 4y = 40\) to the second equation \(-12x + 4y = 30\): \[ (12x - 4y) + (-12x + 4y) = 40 + 30 \]This simplifies to: \[ 0 = 70 \] Since this is a contradiction, the system has no solution.

Step 5: Conclusion of the System

Because the addition of the two equations results in a contradiction \(0 = 70\), the system of equations is inconsistent, indicating that there is no solution.

Key Concepts

Inconsistent System

When dealing with a system of linear equations, an inconsistent system occurs when there is no set of values that satisfies all equations simultaneously. This means the system does not have a solution. You will often recognize an inconsistent system when you end up with a contradiction like \(0 = 70\). This result can occur after performing operations such as elimination on the equations.
In the case of our example, once we simplified the equations:

• We ended up with two straight lines that do not intersect.
• If graphed, these lines would be parallel, indicating there's no common solution point.
• You must always look out for contradictions as signs of inconsistency.

Understanding inconsistent systems is crucial in math as it tells us whether a problem can be solvable with the given conditions.

Elimination Method

The Elimination Method is a powerful tool used to solve systems of equations. It involves adding or subtracting equations to eliminate one of the variables, simplifying the process. Here is how you typically perform it:

• First, align coefficients for one of the variables by multiplication.
• Next, add or subtract the equations to eliminate that variable.
• Finally, solve for the remaining variable.

In our example, we multiplied the first equation by 2 to adjust the coefficients of \(y\) to match those in the second equation. This setup allowed us to add the two resulting equations, effectively canceling \(y\) out. The aim is to create a simpler equation involving only \(x\) or \(y\), thus making it easier to solve.
The Elimination Method requires careful preparation and is especially useful when coefficients can be adjusted to easily cancel out terms.

Decimal Coefficients

Decimal coefficients can make solving linear equations more complex because they require precise calculations and can complicate the process. To simplify equations with decimals, we can use a method known as scaling.
In our exercise, both equations had decimal coefficients. We eliminated the decimals by multiplying the entire equation by 10. This gives us integers, which are easier to work with without losing accuracy.

• Using integers makes it more straightforward to perform operations like addition or subtraction.
• Scaling by multiples of 10 is an effective method to clear decimals.

This process is crucial because it helps avoid errors that can arise from handling decimal operations in each step. A clear calculation with integers usually leads to a less error-prone solution path, making it a smart strategy in solving equations efficiently.