Problem 47 - Precalculus Mathematics 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 6. $$\left\\{\begin{aligned}\frac{1}{3} x-\frac{1}{4} y &=2 \\\\-8 x+6 y &=10\end{aligned}\right.$$

Step-by-Step Solution

Answer

The system has no solution.

Step 1: Eliminate fractions

Starting with the equation \( \frac{1}{3}x - \frac{1}{4}y = 2 \), we eliminate the fractions by finding a common denominator. The common denominator of 3 and 4 is 12. Multiply every term by 12 to get: \( 4x - 3y = 24 \).

Step 2: Set up the rewritten system

With the equation \( 4x - 3y = 24 \) from Step 1, and the second original equation "`-8x + 6y = 10`", we have the system:\[\begin{aligned} 4x - 3y &= 24, \-8x + 6y &= 10. \end{aligned}\]

Step 3: Simplify and compare equations

Notice that the second equation \( -8x + 6y = 10 \) can be simplified by dividing every term by 2 to get \( -4x + 3y = 5 \). Now compare it with the first equation: \( 4x - 3y = 24 \).

Step 4: Add the Equations

Add the equations: \( (4x - 3y) + (-4x + 3y) = 24 + 5 \). This results in \( 0 = 29 \).

Step 5: Analyze the result

Since the equality \( 0 = 29 \) is false, this indicates that the system of equations has no solution.

Key Concepts

Elimination Method

The elimination method is a powerful technique for solving systems of equations. It involves removing one variable by cleverly adding or subtracting equations. This way, you're left with a simpler equation to solve.

Here's how it often works:

• Choose one variable to eliminate between the equations by aligning them. In this example, aligning the coefficients of the variables was a helpful start.
• Add or subtract equations to eliminate that variable. Sometimes you might need to multiply equations by constants first to align terms for elimination.

In the original step-by-step solution, the subtraction of aligned equations results in an equation like "0 = 29". This is a key indicator that the system cannot be solved using this method because it leads to a contradiction.

When you see something this illogical, it quickly tells you there’s no way to balance both equations simultaneously. Thus, while the elimination approach is usually straightforward, you might run into scenarios indicating something else, like no possible solution.

Fractions in Equations

Fractions in equations can make solving them seem complex. However, eliminating fractions is a handy first step to simplify things.

Start by finding a common denominator among the fractions. Multiply every term of the equation by this common denominator.

For example, the given problem had fractions in the first equation, with denominators 3 and 4. The least common denominator is 12. By multiplying the whole equation by 12, the fractions disappear, making it much easier to continue solving the problem efficiently.

After removing fractions, the equation becomes simpler, with whole numbers instead of fractions, allowing you to see more clearly the relationships between variable terms. This clarity is crucial when you're solving using techniques like elimination or substitution.

No Solution Scenario

Sometimes, when solving a system of equations, you reach a situation where there is no solution. This often happens when both equations represent parallel lines that never intersect.

In the example provided, after eliminating the fractions and simplifying the equations through arithmetic manipulation, we end up with a non-intersecting scenario, which is evidenced by the equation "0 = 29".

Remember, a result like this, where you have an inherently false statement, hints at the possibility of two parallel lines or entirely inconsistent equations—meaning there's no possible value that satisfies both equations at once.

Recognizing this scenario is important because it saves you from making endless calculations, helping you understand that sometimes, the system you’re dealing with simply has no solution.