Problem 37 - A Graphical Approach to Precalculus with Limits

Question

Solve each system by elimination. $$\begin{aligned}x-y &=9 \\\\-2 x+2 y &=-18\end{aligned}$$

Step-by-Step Solution

Answer

The system has infinitely many solutions.

Step 1: Align the Equations

Write down the given system of equations clearly. The system is: \[x - y = 9 \]\[-2x + 2y = -18 \]

Step 2: Multiply the First Equation

To eliminate the variable \( y \), make the coefficients of \( y \) in both equations equal in magnitude but opposite in sign. Since the second equation has a coefficient of \( 2 \) for \( y \), multiply the entire first equation by 2. The first equation becomes: \[2(x - y) = 2(9)\]\[2x - 2y = 18\]

Step 3: Add the Equations

Add both equations together to eliminate \( y \):\[2x - 2y + (-2x + 2y) = 18 + (-18)\]This simplifies to: \[0 = 0\]

Step 4: Analyze the Result

The result \( 0 = 0 \) indicates that the two equations are dependent and represent the same line. This means there are infinitely many solutions.

Key Concepts

Elimination Method

The elimination method is a powerful technique for solving systems of linear equations. The main idea behind this method is to eliminate one of the variables by adding or subtracting equations, which results in a simpler equation. Here's a quick rundown of how it works:

• First, align the equations so that they are easy to compare with respect to each variable.
• Next, transform one or both equations so the coefficients of one of the variables are equal in magnitude but opposite in sign. This often involves multiplying an entire equation by a constant.
• Once the coefficients match, add or subtract the equations to eliminate that variable.
• With one variable gone, solve the remaining single variable equation.
• Finally, substitute back to find the value of the eliminated variable.

In our exercise, we multiplied the first equation by 2 to align the coefficients of \( y \) with opposite signs and then added them, effectively removing \( y \) and simplifying the system.

Dependent System

A dependent system of equations occurs when two or more equations in a system are not independent. This means the equations describe the same line or plane, sharing all their points. Therefore, they do not present new information but rather restate the same relationship in different forms.Here’s how you can tell a system is dependent:

• When simplified, the equations result in identical or scalar multiples of each other.
• The elimination method results in a trivial equality, like \( 0 = 0 \).

If this happens, it means the equations overlap completely, resulting in multiple solutions rather than a unique intersection point.

Infinitely Many Solutions

When a system of equations has infinitely many solutions, it means every point on the line that one equation represents is also on the line of the other equation. Essentially, the two equations describe the same geometric object in space.You can identify such systems as follows:

• After elimination or substitution, if you are left with an identity (like \( 0 = 0 \)), this indicates that the two equations are the same.
• This suggests they overlap perfectly and every solution of one is a solution to the other.

This concept plays out in our example where both equations simplify down to the identity \( 0 = 0 \), affirming that they are the same line with all points in common. Hence, there are infinitely many solutions.