Problem 71 - 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. $$ \begin{aligned} &x+y=20\\\ &x-y=8 \end{aligned} $$

Step-by-Step Solution

Answer

The system is consistent and independent with solution \((x, y) = (14, 6)\).

Step 1: Align the Equations

We are given the system of equations: \( x + y = 20 \) and \( x - y = 8 \). We will start by rewriting both equations in a similar layout for ease of elimination.

Step 2: Add the Equations

To eliminate one of the variables, add the two equations: \((x + y) + (x - y) = 20 + 8\). Notice that adding these will eliminate \(y\).

Step 3: Simplify the Sum

After adding, the equation simplifies to: \(2x = 28\). The terms with \(+y\) and \(-y\) cancel each other out.

Step 4: Solve for x

Divide both sides of the equation \(2x = 28\) by 2 to isolate \(x\). Thus, \(x = 14\).

Step 5: Substitute to find y

Select one of the original equations (e.g., \(x + y = 20\)) and substitute \(x = 14\) into it. This gives \(14 + y = 20\).

Step 6: Simplify to Find y

Solve for \(y\) by subtracting 14 from both sides: \(y = 20 - 14 = 6\).

Step 7: Verify the Solution

Substitute \(x = 14\) and \(y = 6\) back into the second equation \(x - y = 8\) to check: \(14 - 6 = 8\). This confirms our solution.

Step 8: Determine Consistency and Dependency

Since we found a solution, the system is consistent. Since the solution is unique, the system is independent.

Key Concepts

Elimination Method

The elimination method is a powerful technique for solving systems of equations. It involves adding or subtracting equations to remove a variable, which simplifies the problem. This method is especially useful when dealing with a pair of linear equations.

To start the elimination process, align your equations so that like terms are vertically stacked. This visual alignment helps you spot opportunities to eliminate variables. In the example of the system \[\begin{aligned} x + y &= 20\ x - y &= 8 \end{aligned} \], adding the equations directly cancels out the \(y\) terms, leaving you with a single equation in terms of \(x\): \[(x + y) + (x - y) = 20 + 8\].

• Step-by-step: Add or subtract the equations.
• Focus on a variable: Target one variable to eliminate first.
• Isolate remaining: Solve for the remaining variable.

Use the result from the simplified equation to find the value of the other variable. Once you have both values, verify by plugging them back into the original equations.

Consistent System

A consistent system of equations is a system that has at least one solution. This means that in the context of two linear equations, the lines represented by these equations intersect at least at one point.

In our example, the initial calculations led us to find that \(x = 14\) and \(y = 6\), which are solutions for both equations. This shows that the system is consistent because the equations do meet at exactly these values for \(x\) and \(y\).

• Unique solution: If there's exactly one solution, the lines intersect at a single point.
• Graphically: A consistent system means the lines are not parallel.

A consistent system is highly valuable because it assures that a solution exists and can be practically applied or interpreted.

Dependent and Independent Equations

Understanding whether a system of equations is dependent or independent is crucial in interpreting the nature of its solutions.

In an independent system, each equation provides unique information, and the lines intersect at exactly one point. In our given system, we determined that there was one solution \((x=14, y=6)\). This indicates the lines cross each other exactly at one point, showing the equations are independent.

• Dependent equations: These represent the same line, leading to infinitely many solutions.
• Independent equations: Represent distinct lines intersecting at one point, like in our problem.

Checking solutions in both equations confirms the system's nature. An independent system being consistent ensures practicality in prediction and solving for variables.