Problem 30 - Precalculus

Question

Solve each system of linear equations by elimination. $$\begin{aligned} &11 x+3 y=3\\\ &22 x+6 y=6 \end{aligned}$$

Step-by-Step Solution

Answer

The system has infinitely many solutions because the equations describe the same line.

Step 1: Analyze the System

We have the system of equations: \( 11x + 3y = 3 \) and \( 22x + 6y = 6 \). Notice that the second equation is twice the first equation. This suggests the two equations are not independent, which means they describe the same line.

Step 2: Multiply Equation to Simplify

Although it seems the equations already hint at redundancy, let's multiply the first equation by 2 for clear comparison: \( 2(11x + 3y = 3) \) which results in \( 22x + 6y = 6 \). This is exactly the second equation.

Step 3: Conclude Elimination Solution

Since both equations are the same after simplification, it shows that they are dependent, meaning the system of equations has infinitely many solutions. Every point \( (x, y) \) that satisfies one equation will satisfy the other.

Key Concepts

Elimination Method

The elimination method is a popular technique used to solve systems of linear equations. This approach aims to eliminate one variable by combining the two equations. By accomplishing this, it simplifies the problem to a single equation with one variable. To effectively use this method:

• Align the equations vertically so that like terms are in the same column.
• Ensure that the coefficients of one of the variables are equivalent in the equations. You might need to multiply one or both equations by a constant to achieve this.
• Subtract or add the equations to eliminate one of the variables, leading to a simpler equation.

In our original exercise, when the first equation is multiplied by 2, it matches exactly with the second equation. This means any attempt to subtract or add would result in zero, revealing that the variables will be eliminated in favor of understanding the relationship between the equations.

Dependent Equations

Dependent equations are a key concept when dealing with systems of linear equations. These are equations that, through manipulation, reveal themselves as essentially identical. Such equations often signify that instead of intersecting at a single point, the lines represented by these equations coincide, implying an overlap.
In the original exercise, after multiplying and simplifying the first equation, it becomes clear that both equations are the same. This realization shows they are dependent because they express the same linear relationship and graph as the same line.
Dependent equations are significant because they indicate the system does not have a unique solution. Instead, it suggests that the possibilities are much broader, linked to the concept of having infinitely many solutions.

Infinitely Many Solutions

When a system of equations has infinitely many solutions, it means that every point on the line represented by the equations is a solution. This occurs typically with dependent equations where each equation expresses the same mathematical relationship as the others.
For a system of linear equations, infinitely many solutions imply that there isn't just one intersection point. Instead, the equations graph as the same line, so every point on the line satisfies both equations simultaneously.
In our case, multiplying the first equation by a constant revealed it as a duplicate of the second. Therefore, all \((x, y)\) pairs that solve one equation will solve both, leading to an infinite number of solutions along the line.

• This often indicates redundancy in data or constraints.
• It signifies that there's no unique solution, but a plethora of solutions forming a line.