Problem 38 - A Graphical Approach to Precalculus with Limits

Question

Solve each system by elimination. $$\begin{aligned}x+y &=4 \\\3 x+3 y &=12\end{aligned}$$

Step-by-Step Solution

Answer

The system has infinitely many solutions as both equations represent the same line.

Step 1: Write Down the Equations

The given system of equations is \[x + y = 4\] and \[3x + 3y = 12.\] These two equations are representing lines in a coordinate plane, and we are tasked with finding their intersection using the elimination method.

Step 2: Simplify the Second Equation

Observe that the second equation \[3x + 3y = 12\] can be simplified by dividing the entire equation by 3. Doing so results in the equation \[x + y = 4.\] This shows that both equations are actually the same line.

Step 3: Analyze Results

Since after simplification both equations are the same: \[x + y = 4,\] this means they have infinitely many solutions as each point on their graph is an intersection point. This situation indicates that the equations are dependent and the system has infinitely many solutions.

Key Concepts

Understanding Systems of Equations

A system of equations is a collection of two or more equations with the same set of unknowns. In our case, the given system consists of two equations:

• \( x + y = 4 \)
• \( 3x + 3y = 12 \)

The goal is to find the values of the unknowns that satisfy all equations simultaneously. Geometrically, this means finding the point(s) where the lines represented by these equations intersect. Different techniques like graphing, substitution, or elimination can be used to find these solutions. The elimination method, which is highlighted here, efficiently removes one variable by adding or subtracting equations, leading to a simpler equation to solve.
By eliminating variables, we can explore whether there is a unique solution, infinitely many solutions, or no solution at all.

Infinite Solutions Explained

When solving a system of equations, you might come across a scenario where the equations simplify to a single equation or a tautology (a statement that is always true). This indicates infinite solutions, which means every point on the line represents a solution to the system.
In this exercise, after simplifying \( 3x + 3y = 12 \) to \( x + y = 4 \), both equations are identical. Therefore, both lines are exactly the same on a coordinate plane.
This means that every point on the line \( x + y = 4 \) is a solution, leading to an infinite number of solutions.
Recognizing infinite solutions is simple: after performing simplifications, if both equations express the same relationship between variables, all points fulfilling this new equation are solutions.

Dependent Equations and What They Mean

Dependent equations are equations where one is a multiple of the other. In essence, they represent the same line in a graph. This dependency means that the system of equations does not form a unique point of intersection because the lines lie atop one another.
In our example, the second equation became \( x + y = 4 \) after simplification, matching the first. This demonstrates that the two equations are dependent.
Dependent equations lead to infinite solutions, as every point on the line where the equations overlap provides a valid solution to the system.
Understanding dependent equations helps in identifying systems that don't offer a unique solution but rather a line of solutions.