Problem 33 - Precalculus: Mathematical for Calculus

Question

Solve the system, or show that it has no solution. If the system has infinitely many solutions, express them in the ordered-pair form given in Example 3. $$\left\\{\begin{aligned} \frac{1}{3} x-\frac{1}{4} y &=2 \\ -8 x+6 y &=10 \end{aligned}\right.$$

Step-by-Step Solution

Answer

The system has no solution.

Step 1: Clear Fractions in the First Equation

To eliminate the fractions, multiply the first equation by 12 (the least common multiple of 3 and 4). \[ 12 \times \left( \frac{1}{3}x - \frac{1}{4}y = 2 \right) \Rightarrow 4x - 3y = 24 \] This simplifies the first equation to \(4x - 3y = 24\).

Step 2: Write Down the Simplified System of Equations

The system is now:\[ \begin{aligned} 4x - 3y &= 24 \ -8x + 6y &= 10\end{aligned} \]

Step 3: Reduce the Second Equation

Divide the second equation by 2 to make calculations easier:\[ -8x + 6y = 10 \Rightarrow -4x + 3y = 5 \]

Step 4: Analyze the System for Consistency

The simplified system of equations is now:\[ \begin{aligned} 4x - 3y &= 24 \ -4x + 3y &= 5\end{aligned} \]Adding these equations together results in:\[ (4x - 4x) + (-3y + 3y) = 24 + 5 \Rightarrow 0 = 29 \]This contradiction (\(0 = 29\)) means the system has no solution.

Key Concepts

Fraction Elimination

When dealing with a system of linear equations, especially those involving fractions, simplifying the equations can make it easier to work with them. Fraction elimination is the process of removing fractions from an equation to simplify it.Here's how it works:

• Identify the least common multiple (LCM) of the denominators in the equation. In our example, the LCM of 3 and 4 is 12.
• Multiply every term of the equation by this LCM to clear the fractions. This turns the equation into one with only whole number coefficients.

For instance, the initial equation \( \frac{1}{3} x - \frac{1}{4} y = 2 \) is transformed into \( 4x - 3y = 24 \) by multiplying each term by 12.Clearing fractions simplifies calculations and helps avoid errors, making it easier to solve or analyze the system of equations.

System Consistency

The concept of system consistency pertains to whether a system of equations has at least one solution. A consistent system means that there are values for the variables that satisfy all equations simultaneously.There are three possible outcomes concerning consistency:

• The system is consistent and has a unique solution. This is often verified by finding distinct values for each variable that work across all equations.
• The system is consistent and has infinitely many solutions. This occurs when the equations describe the same line or plane, leading to multiple solutions, often expressed in parameterized form.
• The system is inconsistent and has no solution. This occurs when the equations contradict each other, such as resulting in a statement like \( 0 = 29 \).

Before solving, assessing the system for consistency helps determine the approach and whether a valid solution is possible.

No Solution

In certain cases, solving a system of linear equations may result in a logical contradiction, indicating the absence of any solution. The occurrence of a contradiction, such as obtaining a false statement like \( 0 = 29 \), illustrates that the system is inconsistent.Key ideas when facing no solution systems:

• The graphs of the equations do not intersect; they are parallel lines in two dimensions.
• For systems of more than two equations, you may find a similar outcome where no common point exists across all dimensions.
• Detecting no solution early can prevent unnecessary calculations. Recognizing contradictory equations, such as when attempting to make both sides of an equation equal during addition or subtraction, is crucial.

An early sign of no solution is having proportional coefficients for corresponding variables in all equations, accompanied by differing constant terms.