Problem 76 - College Algebra with Modeling and Visualization

Question

Use elimination to solve the system of equations, if possible. Identify the system as consistent or inconsistent. If the system is consistent, state whether the equations are dependent or independent. Support your results graphically or numerically. $$ \begin{aligned} &2 x+3 y=5\\\ &5 x-2 y=3 \end{aligned} $$

Step-by-Step Solution

Answer

The system is consistent and independent with solution \((x, y) = (1, 1)\).

Step 1: Review the System of Equations

We have two equations: (1) \(2x + 3y = 5\) and (2) \(5x - 2y = 3\). Our task is to solve for \(x\) and \(y\) using the elimination method and determine whether the system is consistent or inconsistent and whether the equations are dependent or independent.

Step 2: Multiply Equations to Facilitate Elimination

To eliminate one variable, we need the coefficients of one variable to be the same (or opposites). Multiply equation (1) by 2 to get \(4x + 6y = 10\) and equation (2) by 3 to get \(15x - 6y = 9\). Now, the \(y\)-terms in both equations have coefficients of \(6\) but with opposite signs.

Step 3: Add the Modified Equations

Add the modified equations to eliminate \(y\): \((4x + 6y) + (15x - 6y) = 10 + 9\), resulting in \(19x = 19\).

Step 4: Solve for x

Divide each side of \(19x = 19\) by 19 to solve for \(x\): \(x = 1\).

Step 5: Substitute to Find y

Substitute \(x = 1\) back into one of the original equations to find \(y\). Using equation (1), \(2(1) + 3y = 5\) simplifies to \(2 + 3y = 5\). Solve for \(y\): \(3y = 3\) so \(y = 1\).

Step 6: Determine Consistency and Independence

The system has a unique solution \((x, y) = (1, 1)\). Thus, it is consistent and the equations are independent since there is exactly one solution.

Step 7: Graphical Support

Graphing both equations on a coordinate plane, they intersect at exactly one point: \((1, 1)\), supporting that the system is consistent and independent.

Key Concepts

Consistent System

A system of equations, like the one in the exercise, can be described as consistent if there is at least one solution that satisfies all the equations simultaneously. In simpler terms, if you can find at least one set of values for the variables that make all equations true, then the system is consistent.

The solution process in the exercise used the elimination method to find that the system has exactly one solution: \(x = 1\) and \(y = 1\). This singular solution confirms the consistency of the system. For consistent systems:

• There can be one solution—like our exercise indicates—making the system not only consistent but also the equations independent.
• Or there can be infinitely many solutions if the equations are dependent and represent the same line.

Visual verification, such as graphing, offers a great way to confirm consistency, as intersecting lines show precisely where solutions exist.

Independent Equations

In a consistent system, as demonstrated, the equations can either be independent or dependent. Equations are independent if they have only one solution, meaning they cross at a single point on a graph.

In the problem from the exercise, after applying the elimination method, we identified a unique point \(x = 1\) and \(y = 1\). This solution indicates that the lines represented by the two equations intersect at exactly one point, confirming that they are independent equations.

• Independent equations form distinct lines on a graph which cross only once.
• If more solutions existed, or if no unique solution could be found, it would suggest that the equations aren’t independent.

The main characteristic of independent equations is their capacity to validate the presence of exactly one solution in a system.

System of Equations

A system of equations consists of multiple equations with multiple variables. Solving these systems requires finding values for the variables that simultaneously satisfy all the equations. The exercise involved a straightforward two-variable system of linear equations:

• \(2x + 3y = 5\)
• \(5x - 2y = 3\)

These equations together form what is known as a **linear system** because each equation graphs as a line in two-dimensional space. Solving such systems involves determining the point(s) where these lines intersect.

Each method for solving systems—such as elimination, substitution, or graphing—offers unique ways of approaching the solution. By using elimination in the exercise, we aligned one of the variable coefficients to make the variables cancel out easily.

Solving a system of equations doesn’t just find a numerical answer—it provides insights about relationships between the variables involved.