Problem 49 - College Algebra with Modeling and Visualization

Question

If possible, solve the system of linear equations and check your answer. $$ \begin{array}{rr} 3 x-2 y= & 5 \\ -6 x+4 y= & -10 \end{array} $$

Step-by-Step Solution

Answer

The system has infinitely many solutions; any point on the line satisfies both equations.

Step 1: Identify the System of Equations

The system of equations given are:\[ 3x - 2y = 5 \] and \[ -6x + 4y = -10 \]

Step 2: Check for Linearity Dependence

Notice that the second equation is a multiple of the first. Multiply the first equation by 2 to see if it matches the second equation. After multiplication, \[ 2(3x - 2y) = 2(5) \] gives \[ 6x - 4y = 10 \], which, when rearranged, results in \[ -6x + 4y = -10 \] being equivalent to the second equation. This indicates that the equations are dependent.

Step 3: Interpret the Results

Since one equation is a multiple of the other, the two equations represent the same line. Therefore, the system has infinitely many solutions, where solutions can be any point on the line.

Step 4: Verify an Example Solution

To verify, take an arbitrary value of \(x\), say \(x = 1\), and substitute it into either equation. Using the first equation: \[ 3(1) - 2y = 5 \Rightarrow 3 - 2y = 5 \Rightarrow -2y = 2 \Rightarrow y = -1 \]. The point \((1, -1)\) is a solution. Substitute into the second equation to verify: \[ -6(1) + 4(-1) = -10 \Rightarrow -6 - 4 = -10 \], which confirms that the point is indeed a solution.

Key Concepts

System of Equations

A system of equations refers to a set of two or more equations that have the same variables. In our original exercise problem, we are working with two equations, which means they form a system:

• Equation 1: \(3x - 2y = 5\)
• Equation 2: \(-6x + 4y = -10\)

A linear equation means that its graph is a straight line, and when we deal with systems of equations, we typically want to find any points that satisfy all equations in the system. These points are where the lines representing each equation intersect.
There are several possible outcomes when solving a system of linear equations:

• One solution: This occurs when the lines intersect at a single point.
• No solution: This happens when the lines are parallel and never intersect.
• Infinitely many solutions: This arises when the lines coincide, representing the same line.

Understanding these outcomes can help in visualizing and properly solving systems of equations.

Independent Solutions

In the context of a system of linear equations, independent solutions occur when each equation in the system provides distinct information, meaning the lines intersect at exactly one point. When we solve such systems, we are generally looking for this point of intersection, which represents the unique solution that satisfies both equations.
For independently solvable systems, you can use several methods:

• Substitution Method: Solve for one variable in terms of the other, and substitute it back into the second equation.
• Elimination Method: Add or subtract equations to eliminate one variable, making it easier to solve for the other.
• Graphical Method: Graph both equations and find the intersection point visually.

In our original example, however, the equations are not independent. As demonstrated, one equation is simply a multiple of the other, which means they coincide and do not intersect at just a single point.

Dependent Equations

Dependent equations occur when one equation in the system is a multiple of the other, leading to them representing the same line. This implies that all the solutions to one equation are solutions to the other as well, resulting in infinitely many solutions. Essentially, each equation provides the same information about the variables.
In our example:

• Equation 1: \(3x - 2y = 5\)
• Equation 2: \(-6x + 4y = -10\)

By multiplying Equation 1 by 2, you obtain Equation 2, confirming they are dependent.
When studying systems with dependent equations, it's crucial to verify if one equation can be derived from the other. Once identified, every point on the line \(3x - 2y = 5\) is a solution because the system describes the exact same line twice. Hence, instead of finding a unique solution, any point on this line could be considered a valid solution, leading to infinitely many solutions. For such systems, establishing one of the dependent equations in terms of a parameter can help describe the complete solution set.