Problem 84 - College Algebra with Modeling and Visualization

Question

Solve the system, if possible. $$ \begin{array}{rr} -\frac{1}{3} x+\frac{1}{6} y= & -1 \\ 2 x-y= & 6 \end{array} $$

Step-by-Step Solution

Answer

Infinitely many solutions.

Step 1: Convert Fractional Coefficients

To eliminate fractions, multiply the entire first equation by 6 to get integer coefficients.First equation becomes: \[ -2x + y = -6 \] Now the system is: \[\begin{align*}-2x + y &= -6 \2x - y &= 6\end{align*}\]

Step 2: Add Equations

Add the two equations together to eliminate \(x\):\[\begin{align*}(-2x + y) + (2x - y) &= -6 + 6 \0 &= 0\end{align*}\]This shows that the equations are dependent and represent the same line. Thus, there are infinitely many solutions.

Key Concepts

Dependent Equations

Dependent equations are a fundamental aspect to understand when dealing with systems of linear equations. Two equations are considered dependent if they illustrate the same line. This means no matter how you solve them, they are essentially equivalent.
This can happen if one equation is a multiple of the other. When graphed, dependent equations will sit directly atop one another, visually showing they are interconnected or overlapping.
A key indicator of dependent equations in a solved system, like the one in the example, is that all variables may cancel out, leading to a statement that's always true, such as 0 = 0. This results from their dependent nature.

Fractional Coefficients

Fractional coefficients can make solving systems of linear equations a bit trickier, but understanding how to handle them is crucial. These arise when the numbers in front of the variables, known as the coefficients, are fractions.

To simplify the handling of fractional coefficients, a common step is to eliminate the fractions. This can be done by multiplying all parts of the equation by the least common multiple (LCM) of the denominators.

• In our example, the equation \(-\frac{1}{3}x + \frac{1}{6}y = -1\) was multiplied by 6, which is the LCM of 3 and 6.
• This conversion transforms the equation into \(-2x + y = -6\), which is much easier to work with.

Handling the equations without fractional coefficients simplifies further calculation and alleviates potential errors in solving.

Infinite Solutions

Understanding infinite solutions in systems of linear equations is crucial. When two equations have infinite solutions, it means that there are countless combinations of values for the variables that satisfy both equations.
Infinite solutions occur in systems where the equations are dependent. In the example system, both equations represent the same line, implying every point on the line is a solution.

• The result obtained, \(0 = 0\), tells us that any pair \((x, y)\) fitting one equation fits the other as well.
• This leads to infinitely many solutions, implying the entire line described is the solution set.

Recognizing infinite solutions helps us understand the nature of the relationship between the equations involved, significantly impacting how we interpret the system.