Problem 22 - Elementary and Intermediate Algebra

Question

Solve each system. If a system’s equations are dependent or if there is no solution, state this. $$\begin{aligned} x-y+z &=4 ,\\\ 5 x+2 y-3 z &=2 ,\\\ 4 x+3 y-4 z &=-2 \end{aligned}$$

Step-by-Step Solution

Answer

The system of equations is dependent, so it has infinitely many solutions.

Step 1 - Write down the system of equations

\[ \begin{aligned} 1. & \ x - y + z &= 4 \ 2. & \ 5x + 2y - 3z &= 2 \ 3. & \ 4x + 3y - 4z &= -2 \end{aligned} \]

Step 2 - Eliminate one variable

Multiply the first equation by -4 and add it to the third equation to eliminate variable x:\[ -4(x - y + z) + (4x + 3y - 4z) = -4(4) + (-2) \]This simplifies to:\[ -4x + 4y - 4z + 4x + 3y - 4z = -16 - 2 \]\[ 7y - 8z = -18 \]Label this equation as equation (4).

Step 3 - Eliminate the same variable from another pair of equations

Multiply the first equation by -5 and add it to the second equation to eliminate variable x:\[ -5(x - y + z) + (5x + 2y - 3z) = -5(4) + (2) \]This simplifies to:\[ -5x + 5y - 5z + 5x + 2y - 3z = -20 + 2 \]\[ 7y - 8z = -18 \]Label this as equation (5).

Step 4 - Observe new system

Note that equations (4) and (5) are identical: \[ 7y - 8z = -18 \]This means that the two equations are dependent.

Step 5 - Determine system classification

Since the second and third equations are dependent, the system of equations is dependent. Therefore, there are infinitely many solutions.

Key Concepts

Understanding Dependent Equations

In a system of equations, 'dependent equations' are those that are essentially the same after some manipulation. This means that one equation can be obtained from another by multiplying, adding, or subtracting. When equations are dependent, they do not provide unique information about the variables. Instead, they represent the same geometric line in space. For example, in our original exercise, after simplifying the equations, we found that the new equations formed were the same. This indicated that the relationship among the variables is constant, leading to dependent equations.

Using the Elimination Method

The 'elimination method' is an algebraic technique used to solve systems of equations. It involves combining equations to eliminate one of the variables, making it easier to solve for the others. Here’s a simple way to understand it:

• First, choose a variable to eliminate.
• Adjust coefficients to align terms.
• Add or subtract equations to cancel out the selected variable.

In our exercise, we chose to eliminate the variable 'x'. By multiplying the first equation by suitable numbers and adding them to the second and third equations, we successfully canceled out 'x'. This reduction simplifies the system and helps us spot dependent equations.

Infinitely Many Solutions

Sometimes, systems of equations do not have a single unique solution. Instead, they may have 'infinitely many solutions'. This occurs when the equations are dependent, forming the same line in a geometric sense. In other words, for every value of one variable, there is a corresponding value of another variable that satisfies both equations. For the given example, once we reduced the system, we observed that two separate equations were identical. This means that any combination of 'y' and 'z' that fits the simplified equation will be a solution to the original system. Thus, there's no single unique solution but infinitely many solutions.