Problem 49 - 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 in the First Equation

To make it easier to work with the first equation, eliminate the fractions by finding a common denominator for the terms. The common denominator of 3 and 4 is 12. Multiply each term in the equation \( \frac{1}{3}x - \frac{1}{4}y = 2 \) by 12 to remove the fractions: \[ 12 \left( \frac{1}{3}x \right) - 12 \left( \frac{1}{4}y \right) = 12 \cdot 2 \]This simplifies to:\[ 4x - 3y = 24 \]

Step 2: Rewrite the System of Equations

Now, rewrite the entire system using the new equation from Step 1:\[\begin{align*}4x - 3y &= 24 \-8x + 6y &= 10\end{align*}\]

Step 3: Multiply to Align Coefficients for Elimination

We can align the coefficients of \( y \) for elimination by multiplying the first equation by 2 to match the coefficient from the second equation:\[\begin{align*}2(4x - 3y) &= 2(24) \-8x + 6y &= 10\end{align*}\]This gives us:\[\begin{align*}8x - 6y &= 48 \-8x + 6y &= 10\end{align*}\]

Step 4: Add the Equations to Eliminate a Variable

Add the two equations from Step 3 to eliminate \( y \):\[(8x - 6y) + (-8x + 6y) = 48 + 10\]This simplifies to:\[ 0 = 58 \]This result is a contradiction, suggesting the equations have inconsistent terms.

Step 5: Determine System Solution Status

Since the last step resulted in a contradiction (\( 0 = 58 \)), it indicates there is no set of \( x \) and \( y \) values that can satisfy both equations simultaneously. Therefore, the system has no solution.

Key Concepts

Solving Linear Equations

Solving linear equations involves finding values for variables that satisfy given equations. In a system of linear equations, like the one in this exercise, you have more than one equation to deal with. You are looking for the values of variables that make all the equations true.
Linear equations can be solved using various methods such as substitution, elimination, or graphing. Each method has its own advantages depending on the situation.
It's essential to transform complex equations into simpler forms to make the solution more manageable. This often involves eliminating fractions, aligning coefficients, or simplifying expressions.

• First, transform any fractions to whole numbers to make calculations easier.
• Next, align coefficients in a way that simplifies the elimination process.

These steps set up everything we need to identify whether the system of equations is consistent, inconsistent, or has infinitely many solutions.

Inconsistent System

An inconsistent system of equations occurs when there is no solution. This means that the lines represented by these equations do not intersect. In a graphical sense, these lines are parallel.
In the step-by-step solution, you saw the contradiction arise when adding the equations, leading to an untrue statement like \(0 = 58\). This contradiction signifies that our system is inconsistent.

• Equations that are inconsistent will never meet; hence, they have no points in common.
• A contradiction in the elimination process, such as one that results in \(0 = 58\), definitively shows inconsistency.

This characteristic is crucial in understanding the relationships between the equations in a system. Recognizing inconsistent systems quickly can save time in the solving process.

Elimination Method

The elimination method is a strategic way to solve systems of equations by removing one variable at a time. This is achieved through addition or subtraction, which facilitates solving for the remaining variables. The goal is to make either the \(x\) or \(y\) terms cancel each other out in the equations.
In our exercise, the first step was manipulating the first equation to eliminate fractions. Then, coefficients were aligned to allow for the straightforward elimination of one variable.

• To eliminate a variable, multiply each equation (if necessary) to have matching coefficients for the targeted variable.
• Once coefficients match, add or subtract the equations to cancel out one of the variables.

In this specific case, aligning \(y\)'s coefficients set the stage for elimination, targeting \(y\) in both equations. The result confirmed the inconsistency when the process led to a contradiction rather than a valid statement.