Problem 54 - Intermediate Algebra

Question

Solve each system, if possible. If a system is inconsistent or if the equations are dependent, state this. $$ \left\\{\begin{array}{l} -x+5 y-7 z=0 \\ 4 x+y-z=0 \\ x+y-4 z=0 \end{array}\right. $$

Step-by-Step Solution

Answer

The system is dependent with solutions in the form \((x, y, z) = (18z, 5z, z)\).

Step 1: Rewrite the Equations

Let's first write the given system of equations:1) \(-x + 5y - 7z = 0\)2) \(4x + y - z = 0\)3) \(x + y - 4z = 0\).

Step 2: Eliminate One Variable

We will eliminate \(x\) from the first two equations. Multiply equation 3) by 4 to match the coefficient of \(x\) in equation 2):- Equation 2: \(4x + y - z = 0\)- Equation 3 (multiplied by 4): \(4x + 4y - 16z = 0\)Subtract equation 3 from equation 2:\(4x + y - z - (4x + 4y - 16z) = 0\)This simplifies to:\(-3y + 15z = 0\) => \(y = 5z\).

Step 3: Substitute and Solve for Another Variable

Using \(y = 5z\), substitute in Equation 1: \(-x + 5(5z) - 7z = 0\)This simplifies to:\(-x + 25z - 7z = 0\)\(-x + 18z = 0\)From which we can find:\(x = 18z\).

Step 4: Simplify and Discover Relationships

We have relationships: \(y = 5z\) and \(x = 18z\). Let's verify with the third equation:Substitute \(x = 18z\) and \(y = 5z\) in \(x + y - 4z = 0\):\(18z + 5z - 4z = 0\)Simplifies to: \(19z = 0\)This confirms the solutions are consistent.

Key Concepts

Linear Equations

Linear equations are a type of equation where the variables appear to the power of 1, such as in the form \(ax + by + cz + = 0\). Each term is either a constant or the product of a constant and a single variable. These equations appear straight when graphed on a two-dimensional plane, known as a line.In the given exercise, we have three linear equations with three different variables \(x, y, z\). Each equation paints a different plane in three-dimensional space.

• The first equation is \(-x + 5y - 7z = 0\)
• The second equation is \(4x + y - z = 0\)
• The third equation is \(x + y - 4z = 0\)

Understanding these linear equations is crucial. They allow us to solve for the values of \(x\), \(y\), and \(z\) that satisfy all equations simultaneously. Passing each equation to a common form lets us use various mathematical methods to address such systems.

Substitution Method

The substitution method is one way to solve a system of equations. It involves solving one of the equations for one variable and then substituting that expression into the other equations. This method helps reduce the number of variables in the equations and finds relationships between them.In the step-by-step solution, we began by solving one of the equations for one variable. We found that \(y = 5z\) from the modified first and second equations, which allowed us to express \(y\) in terms of \(z\).Here’s a simple way to visualize the process:

• Choose one equation and solve for a variable (we chose to solve for \(y\)).
• Substitute this value into the other equations to simplify them (substitute \(y = 5z\) into the first equation).
• Continue simplifying and substituting until you solve for all variables.

The substitution method is particularly helpful because it reveals the nature of solutions without the need for more complex operations.

Consistent Systems

A system of equations is called "consistent" if there is at least one set of variable values that satisfies all equations simultaneously. In other words, all equations "agree" with each other for some values of their variables.After applying the substitution method to our system, we determined that the equations led to specific relationships:\(y = 5z\) and \(x = 18z\). Substituting these into one of the original equations ensured that all were satisfied without any contradictions, confirming the system is consistent.Characteristics of a consistent system include:

• Having at least one solution that satisfies all equations.
• Graphically, this means the lines or planes intersect at one or many points.
• Variable values can be specifically determined unless the system is also dependent.

Consistency is crucial when you are solving systems because it tells you that a solution exists. It's the first check in solving any system of equations.

Dependent Equations

Dependent equations occur when the equations are multiples of each other or combine to represent the same line or plane in space. As a result, they do not affect the number of solutions to the system.In the exercise, though we determined that the system is consistent, it also appears to be dependent. This is shown by how all variables are expressed in terms of \(z\) alone. Since \(x = 18z\), \(y = 5z\), and substituting these into any of the original equations results in an identity (like \(19z = 0\)), these equations are dependent.Key points to remember about dependent equations:

• They yield infinitely many solutions when consistent, as one parameter (like \(z\)) can take any value.
• This is characterized by the equations not providing additional unique information about the solution set.
• Graphically, the lines or planes coincide.

Such dependency indicates that solutions exist along a line or plane rather than at a single point, which helps in understanding the system's geometric nature.