Problem 73 - 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. $$ \begin{array}{l} x+3 y=10 \\ x-2 y=-5 \end{array} $$

Step-by-Step Solution

Answer

The system has a unique solution: \( x = 1 \), \( y = 3 \); it is consistent and independent.

Step 1: Write the System of Equations

We have the system of equations: 1. \( x + 3y = 10 \) 2. \( x - 2y = -5 \).

Step 2: Eliminate x by Subtracting

To eliminate \( x \), subtract the second equation from the first: \( (x + 3y) - (x - 2y) = 10 - (-5) \). This simplifies to: \( 5y = 15 \).

Step 3: Solve for y

Divide each side of the equation \( 5y = 15 \) by 5 to solve for \( y \): \( y = 3 \).

Step 4: Substitute y back to find x

Substitute \( y = 3 \) into the first equation \( x + 3y = 10 \): \( x + 3(3) = 10 \). Simplify to \( x + 9 = 10 \). Subtract 9 from both sides to find \( x \): \( x = 1 \).

Step 5: State the Solution of the System

The solution to the system is \( x = 1 \) and \( y = 3 \). The pair \((1, 3)\) satisfies both equations.

Step 6: Determine Consistency and Dependency

Since there is a unique solution, the system is consistent and the equations are independent.

Key Concepts

Systems of Equations

A system of equations consists of two or more algebraic equations involving the same set of variables. The goal when solving these systems is to find values for the variables that satisfy all the equations simultaneously. For instance, consider our system:

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

Here, we're looking for values of \( x \) and \( y \) that make both equations true at the same time. There are different methods to solve systems of equations, such as substitution, graphing, and elimination. In this case, we are using the elimination method to solve the system, which involves adding or subtracting equations to eliminate one of the variables. By doing this, we reduce the system to a single equation with one variable, which is simpler to solve. Once we find the value of one variable, we substitute it back into one of the original equations to find the value of the other variable.

Consistent Systems

A consistent system of equations is one that has at least one solution. This type of system can either have a unique solution or infinitely many solutions. In the given problem, the solution we found is

• \( x = 1 \)
• \( y = 3 \)

This unique solution indicates that the system is consistent. If two lines representing equations intersect at a single point on a graph, the system is consistent with a unique solution. This contrasts with inconsistent systems, where the lines do not meet, meaning there is no common solution that satisfies both equations. Graphically, an inconsistent system would appear as parallel lines that never intersect.

Independent Equations

In a system of equations, independent equations imply that there is exactly one solution for the system. Each equation represents a unique line on a graph that intersects at one point. In the original exercise, solving the system using the elimination method led us to the unique solution

• (\( x = 1, y = 3 \))

This indicates that the equations are independent. When equations are dependent, they represent the same line, or one is a multiple of the other, resulting in infinitely many solutions because they overlap entirely. Here, however, the fact that we have found only one specific solution confirms that each equation is independent and not reliant on the other for its expression.