Problem 79 - 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. $$ \begin{array}{r} 2 x+3 y=2 \\ x-2 y=-5 \end{array} $$

Step-by-Step Solution

Answer

The system is consistent and independent with solution \(\left(-\frac{11}{7}, \frac{12}{7}\right)\).

Step 1: Write down the system of equations

We have the following system of equations to solve using the elimination method: 1. \( 2x + 3y = 2 \) 2. \( x - 2y = -5 \).

Step 2: Eliminate one variable

To eliminate one variable, we can manipulate the equations to cancel out one of the variables. Let's eliminate \(x\). To do this, multiply the second equation by 2:\[(2)(x - 2y) = (2)(-5)\]This gives:3. \(2x - 4y = -10\).

Step 3: Subtract the equations

Now subtract the new equation (Equation 3) from Equation 1 to eliminate \(x\):\[(2x + 3y) - (2x - 4y) = 2 - (-10)\]This simplifies to:\[7y = 12\].

Step 4: Solve for the eliminated variable

Solve for \(y\) by dividing both sides of the equation by 7:\[y = \frac{12}{7}\].

Step 5: Substitute back to find the other variable

Substitute \(y = \frac{12}{7}\) back into one of the original equations to solve for \(x\). Using Equation 2:\[x - 2\left(\frac{12}{7}\right) = -5\]This simplifies to:\[x - \frac{24}{7} = -5\],Solving for \(x\):\[x = -5 + \frac{24}{7} = -\frac{35}{7} + \frac{24}{7} = -\frac{11}{7}\].

Step 6: Determine the consistency and dependency of the system

The solution \((x, y) = \left(-\frac{11}{7}, \frac{12}{7}\right)\) satisfies both equations, indicating the system is consistent. Since there is a unique solution, the equations are independent.

Key Concepts

System of Equations

A system of equations is simply a set of two or more equations that share the same variables. Solving a system of equations involves finding the values of the variables that satisfy all of the equations simultaneously. In our original exercise, we deal with two equations:

• \( 2x + 3y = 2 \)
• \( x - 2y = -5 \)

This means we are looking for specific values of \( x \) and \( y \) that make both equations true at the same time.
The elimination method, used here, allows us to focus on one variable at a time by canceling out the other. This is achieved through addition or subtraction of equations after manipulating them. It simplifies the problem, making it easier to solve for the variables.

Consistent and Inconsistent Systems

When dealing with a system of equations, it's crucial to determine whether the system is consistent or inconsistent:

• Consistent System: A consistent system has at least one set of solutions. This can include having exactly one solution or infinitely many solutions.
• Inconsistent System: An inconsistent system has no solutions at all. The equations would represent parallel lines that never intersect, for instance.

In our case, the system ended up being consistent because we found a unique solution \( (x, y) = \left(-\frac{11}{7}, \frac{12}{7}\right) \). If the equations had no points of intersection or the variables couldn't satisfy both equations, the system would be inconsistent.

Dependent and Independent Equations

Once we establish that a system is consistent, we can further classify it based on whether the equations are dependent or independent:

• Independent Equations: If there is just one unique solution, as seen in our system, the equations are independent. This is often visualized as two intersecting lines at one distinct point.
• Dependent Equations: These occur when the equations are essentially the same or multiples of one another, leading to an infinite number of solutions. Graphically, this would mean the lines coincide.

In the original solution, each equation intersected at just one point, indicating the system of equations was independent. This independence is confirmed analytically with the unique solution we found for \( x \) and \( y \).