Problem 8 - Precalculus

Question

In Exercises \(1-8\), take a trip down memory lane and solve the given system using substitution and/or elimination. Classify each system as consistent independent, consistent dependent, or inconsistent. Check your answers both algebraically and graphically. $$ \left\\{\begin{aligned} \frac{5}{6} x+\frac{5}{3} y &=-\frac{7}{3} \\ -\frac{10}{3} x-\frac{20}{3} y &=10 \end{aligned}\right. $$

Step-by-Step Solution

Answer

The system is consistent dependent with infinitely many solutions as the equations are multiples of each other.

Step 1: Simplify the Equations

First, we simplify both equations to make them easier to work with. The system of equations is: \[\begin{align*}\frac{5}{6}x + \frac{5}{3}y &= -\frac{7}{3} \quad (1)\-\frac{10}{3}x - \frac{20}{3}y &= 10 \quad (2)\end{align*}\]Equation (1) can be multiplied by 6 to eliminate fractions:\[ 5x + 10y = -14 \]Equation (2) can be multiplied by 3 to eliminate fractions:\[ -10x - 20y = 30 \]

Step 2: Choose an Elimination Method

Notice that the coefficients of \( y \) can be aligned for elimination by adding equations together:\[\begin{align*}5x + 10y &= -14 \ -10x - 20y &= 30\end{align*}\]Since the second equation is exactly \(-2\) times the first, these equations represent parallel lines. We can conclude the lines are consistent dependent—they are the same line repeated.

Step 3: Determine Solution Classification

Because one equation is the exact multiple of the other, the system represents dependent equations. Therefore, the system is classified as consistent dependent.

Step 4: Verify Graphically and Algebraically

Algebraically, the system can be shown as having infinitely many solutions through multiplication and manipulation. Graphically, it can be verified by plotting both equations and noticing they coincide (overlap perfectly).

Key Concepts

Substitution Method

The substitution method is a popular technique for solving systems of linear equations. It involves solving one of the equations for one variable in terms of the other variable. Once you do that, you substitute this expression into the other equation. This step-by-step approach simplifies the system by effectively reducing two equations with two variables to a single equation with one variable.

For example, if you have a system like this one:

• Equation 1: \( rac{5}{6}x + rac{5}{3}y = -rac{7}{3} \)
• Equation 2: \( -rac{10}{3}x - rac{20}{3}y = 10 \)

First, solve one of the equations for a single variable, say \( x \), then substitute it into the other equation. This simplifies the process and leads to finding the variable values efficiently. However, here, due to the nature of the equations, you may find after substitution that both result in equivalent equations, indicating dependent lines.

This method works best when one variable can be easily isolated. In some cases, though, especially with fractional coefficients, substitution can become cumbersome. Thus, analyzing before blindly using the method is beneficial.

Elimination Method

The elimination method is another robust approach to solving systems of equations. This method focuses on eliminating one of the variables by algebraically combining the two equations. This can be done by adding or subtracting the equations after aligning their coefficients appropriately.

In our problem, the equations:

• \(5x + 10y = -14\)
• \(-10x - 20y = 30\)

have been manipulated to align the coefficients for elimination. By noticing that the second equation is an exact multiple of the first, one can add or subtract the equations to eliminate either \(x\) or \(y\).

In our case, adding the equations could directly show that they cancel each other out to zero, leading to discovering the dependent system nature. This elimination method is preferred for cases with clearly alignable coefficients and is usually neat and quick for such instances.

Consistent Dependent Systems

Systems are categorized based on their solutions. A consistent dependent system is one where the equations describe the same line, graphically overlapping completely.

In the given exercise, the system:

• \(5x + 10y = -14\)
• \(-10x - 20y = 30\)

represents dependent equations. The second equation can be achieved by multiplying the first through an entire constant factor. This nature of the relationship indicates infinitely many solutions, characteristics of consistent dependent systems.

These systems do not have a single solution but rather a line representing all possible solutions. Graphically, you will see one line rather than two intersecting lines or divergent lines, confirming this consistency. This discovery is vital as it redirects the approach from looking for one solution to understanding the overarching pattern.

Graphical Solution Method

The graphical solution method involves plotting each equation on the same graph to visually inspect the nature of the system. It is a practical method to see if the equations represent intersecting lines, parallel lines, or the same line.

For our equations:

• \(5x + 10y = -14\)
• \(-10x - 20y = 30\)

plotting these would reveal they are the same line. This confirmation is helpful as it visually represents what the algebra has already suggested—consistent dependent lines.

Using a graphing tool or software, plot each line and see their overlap. They will completely coincide, proving infinite solutions. The graph as such provides a clear image, helpful for intuitive understanding, especially if algebra becomes dense or complex in other scenarios. Being able to switch between algebraic and graphical methods enriches problem-solving skills immensely.