Problem 33 - Algebra and Trigonometry

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 of equations has no solution.

Step 1: Eliminate Fractions

First, we eliminate fractions in the equation \(\frac{1}{3} x - \frac{1}{4} y = 2\). Multiply the entire equation by 12 (the least common multiple of 3 and 4). This gives \(4x - 3y = 24\).

Step 2: Write the System of Equations

After eliminating fractions, rewrite the system of equations as follows:\[\begin{aligned} 4x - 3y &= 24 \-8x + 6y &= 10 \end{aligned}\]

Step 3: Simplify the Second Equation

Notice that the coefficients in the second equation can be simplified. Divide the entire equation \(-8x + 6y = 10\) by 2 to simplify it to \(-4x + 3y = 5\).

Step 4: Add the Equations

Add the simplified second equation \(-4x + 3y = 5\) to the first equation \(4x - 3y = 24\) to eliminate \(x\) and \(y\):\[ (4x - 3y) + (-4x + 3y) = 24 + 5 \]This results in \(0 = 29\), which is a contradiction.

Step 5: Conclude No Solution

The equation \(0 = 29\) is a contradiction, indicating the original system of equations is inconsistent and has no solution.

Key Concepts

Elimination Method

The elimination method is a popular technique for solving a system of linear equations. It involves eliminating one variable by adding or subtracting equations, allowing you to solve for the other variable. In our example, we start by adjusting the equations to make eliminating a variable straightforward. After simplifying, you add or subtract the equations from one another such that one of the variables is removed. This method can be especially helpful when the coefficients of one of the variables are already opposites or can be easily manipulated to become opposites. When simplified solutions reveal contradictions, they may indicate the system is inconsistent, which means it has no solution.

Key steps for using the elimination method:

• Ensure the coefficients of one variable are opposites or can be manipulated to become opposites.
• Add or subtract the equations to eliminate that variable.
• Solve the resulting single-variable equation if possible.

Exercise caution with each step to avoid errors that may lead to incorrect conclusions.

Inconsistent System

An inconsistent system of equations means there's no set of values for the variables that will satisfy all equations simultaneously. In simpler terms, the equations represent parallel lines that never intersect, thus having no common point or solution.

In our exercise, after eliminating one variable using the elimination method, we ended up with the statement \( 0 = 29 \), which is clearly false. This indicates a contradiction, thereby confirming the system is inconsistent. Recognizing an inconsistent system is crucial, as it tells you to stop seeking solutions of an equation because there are none.

• If your result is a false statement, such as \( 0 = 29 \), the system is inconsistent.
• This implies the lines represented by the equations do not cross each other.
• No solutions exist for the given equations.

Understanding the signs of an inconsistent system helps prevent wasting time looking for non-existent solutions.

Fractions in Equations

Fractions in linear equations can sometimes complicate calculations. To simplify solving, it is often beneficial to eliminate fractions by multiplying all terms by the least common multiple (LCM) of the denominators. Doing so converts the fractions into whole numbers, making subsequent algebraic manipulation more straightforward.

In solving equations, following these steps when dealing with fractions can help:

• Identify the least common multiple of all denominators in the equations.
• Multiply every term in the equation by this LCM.
• Rewrite the equation as one involving whole numbers only.

In our example, the equation \( \frac{1}{3} x - \frac{1}{4} y = 2 \) was converted to \( 4x - 3y = 24 \) by multiplying the entire equation by 12, the LCM of 3 and 4. This process made the elimination method easier to apply, illustrating the utility of simplifying equations by eliminating fractions.