Problem 48 - Intermediate Algebra

Question

Solve each system by any method, if possible. If a system is inconsistent or if the equations are dependent, state this. $$ \left\\{\begin{array}{l} \frac{x}{2}-\frac{y}{3}=-4 \\ 0.009 x+0.002 y=0 \end{array}\right. $$

Step-by-Step Solution

Answer

The solution is \((x, y) = (-2, 9)\).

Step 1: Clear the Fractions

Multiply the first equation by 6 to eliminate the fractions: \(6 \times \left(\frac{x}{2} - \frac{y}{3}\right) = 6 \times (-4)\). This simplifies to: \(3x - 2y = -24\).

Step 2: Express One Variable in Terms of the Other

Use the second equation \(0.009x + 0.002y = 0\). Express \(y\) in terms of \(x\): \(0.002y = -0.009x\) or \(y = -\frac{0.009}{0.002}x = -4.5x\).

Step 3: Substitute into the First Equation

Substitute \(y = -4.5x\) into the equation \(3x - 2y = -24\). This gives \(3x - 2(-4.5x) = -24\), which simplifies to \(3x + 9x = -24\) or \(12x = -24\).

Step 4: Solve for x

Solve \(12x = -24\) by dividing both sides by 12: \(x = \frac{-24}{12} = -2\).

Step 5: Solve for y

Substitute \(x = -2\) back into \(y = -4.5x\): \(y = -4.5(-2) = 9\).

Step 6: State the Solution

The solution to the system of equations is \((x, y) = (-2, 9)\).

Key Concepts

Consistent and Inconsistent Systems

A system of equations can be classified based on the type of solutions it has. This classification falls into two categories: consistent and inconsistent systems.
The given system of equations is solved without any contradictions, meaning it has at least one set of solutions. This makes it a consistent system. Whereas, an inconsistent system is one that has no possible solution. This happens when the equations represent parallel lines that never intersect, so they have no points in common.
Consistent systems can be further divided into two types:

• Independent: Has exactly one solution.
• Dependent: Has infinitely many solutions.

In our example, since we found a unique solution \((x, y) = (-2, 9)\), the system is consistent and independent.

Dependent Equations

Dependent equations are when two or more equations result in the same line when graphed. This means they have infinite solutions in common. They cannot be solved independently.
To identify if equations are dependent, we look at multiples or manipulations that reduce one equation to the other. In our example, no such case exists as the transformations of the equations do not reduce one completely to another.
This is why our given system is not one of dependent equations. They intersect at only one point, which makes them independent, as noted previously.

Elimination Method

The elimination method is a handy technique to solve systems of equations. It focuses on eliminating one variable by adding or subtracting equations.
This problem primarily uses substitution rather than elimination, but in similar contexts, elimination can reduce complexity:

• First, align equations so that the coefficients of one variable are opposites.
• Add or subtract the equations to eliminate that variable, simplifying to one equation with one unknown.
• Solve this simpler equation, then back-solve for the eliminated variable.

While substitution was shown, elimination is an alternative method, which provides simplicity for certain problems, particularly those not needing fractions to be cleared first.

Fractions in Equations

Fractions in equations can often complicate solving systems because performing arithmetic becomes cumbersome.
To handle fractions effectively, first eliminate them by multiplying the entire equation by the least common multiple of the denominators. This step simplifies arithmetic and keeps calculations clean.
In the exercise, multiplying the first equation by 6 removed fractions due to denominators 2 and 3. This resulted in cleaner and simpler numbers, \(3x - 2y = -24\), making subsequent steps straightforward and more manageable. Handling fractions early on helps in managing the workload in complex algebraic methods.