Problem 10 - Precalculus: Mathematical 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 3. $$\left\\{\begin{array}{c} 3 x+2 y=0 \\ -x-2 y=8 \end{array}\right.$$

Step-by-Step Solution

Answer

The system has a unique solution: \((4, -6)\).

Step 1: Label the Equations

We start by labeling the given system of linear equations for clarity: \( 3x + 2y = 0 \) is Equation 1 and \( -x - 2y = 8 \) is Equation 2.

Step 2: Eliminate One Variable

We will add Equation 1 and Equation 2 to eliminate the variable \( y \). By adding the equations, we get: \[ 3x + 2y + (-x - 2y) = 0 + 8 \] This simplifies to \( 2x = 8 \).

Step 3: Solve for x

Solve the simplified equation from Step 2: \[ 2x = 8 \] Divide both sides by 2 to find \( x \):\[ x = 4 \]

Step 4: Substitute to Find y

Substitute \( x = 4 \) back into either Equation 1 or Equation 2. We use Equation 1: \[ 3(4) + 2y = 0 \] This simplifies to \( 12 + 2y = 0 \). Subtract 12 from both sides to solve for \( y \):\[ 2y = -12 \] Divide by 2:\[ y = -6 \]

Step 5: Present the Solution

The solution to the system is the ordered pair \((x, y) = (4, -6)\). The system has a unique solution because we found specific values for both \(x\) and \(y\).

Key Concepts

Unique Solution

When solving a system of linear equations, a **unique solution** means there is exactly one set of values for the variables that satisfies all the equations simultaneously. In other words, the equations intersect at a single point on a graph.
In our example system:

• Equation 1: \(3x + 2y = 0\)
• Equation 2: \(-x - 2y = 8\)

By solving these equations, we found specific values: \(x = 4\) and \(y = -6\).
This single ordered pair \((4, -6)\) is the point where both equations intersect, indicating a unique solution. If there were no point of intersection, the system would have no solution. Conversely, if the lines were overlapping, there would be infinitely many solutions. However, in this case, with definite values for \(x\) and \(y\), we know the solution is unique.

Variable Elimination

The method of **variable elimination** is a powerful tool for solving systems of linear equations. The goal is to remove one of the variables, making it easier to solve for the remaining one. In our system, we used this method to eliminate the variable \(y\).
Here's how it works in our example:1. We added Equation 1 \(3x + 2y = 0\) to Equation 2 \(-x - 2y = 8\) in order to eliminate \(y\):\[3x + 2y + (-x - 2y) = 0 + 8\]2. The \(2y\) and \(-2y\) terms cancel each other out, simplifying to \(2x = 8\).
By using addition in this way, one variable is eliminated, leaving an equation with a single variable \(x\) to solve.This systematic approach allows for clear and straightforward solutions by reducing complexity and tackling each equation methodically.

Ordered Pair Solution

Once a system of linear equations is solved, the solution is often expressed as an **ordered pair**, representing the coordinate on a graph where the two lines intersect.
In our problem, after using variable elimination and solving for both variables, we obtained the ordered pair solution \((4, -6)\).

• The first number in the ordered pair, \(4\), is the solution for \(x\).
• The second number, \(-6\), is the solution for \(y\).

This ordered pair provides a clear answer to the system.
It's the precise location that satisfies both linear equations. Essentially, it indicates that at \(x=4\) and \(y=-6\), both equations hold true, confirming their intersection point on a coordinate plane.