Problem 12 - Intermediate Algebra

Question

Solve each of the following systems. If the solution set is \(\varnothing\) or if it contains infinitely many solutions, then so indicate. $$ \left(\begin{array}{r} 2 x-y+3 z=1 \\ 4 x+7 y-z=7 \\ x+4 y-2 z=3 \end{array}\right) $$

Step-by-Step Solution

Answer

The system has infinitely many solutions.

Step 1: Choose a Variable to Eliminate

We'll start by eliminating a variable from two pairs of equations. Let's aim to eliminate variable \(x\) first. We'll begin by multiplying Equation 3 by 2 to match the coefficient with \(x\) in Equation 1.

Step 2: Eliminate \(x\) from Equations 1 and 3

Multiply Equation 3 by 2 to get: \(2x + 8y - 4z = 6\). Now subtract Equation 1 from this: \[(2x + 8y - 4z) - (2x - y + 3z) = 6 - 1\]This simplifies to the new equation: \(9y - 7z = 5\). Let's label this as Equation 4.

Step 3: Eliminate \(x\) from Equations 2 and 3

Multiply Equation 3 by 4 to get: \(4x + 16y - 8z = 12\). Subtract Equation 2 from this: \[(4x + 16y - 8z) - (4x + 7y - z) = 12 - 7\]This simplifies to another equation: \(9y - 7z = 5\). Observe that this is identical to Equation 4, indicating a dependency.

Step 4: Simplify the System of Equations

Since Equation 4 is the same as the equation derived in Step 3, we have one valid equation \(9y - 7z = 5\), and two equations (Equation 4) derived from removing \(x\). We no longer need the third equation essentially.

Step 5: Solve for One Variable in Terms of Another

Express \(z\) in terms of \(y\) from Equation 4: \[7z = 9y - 5\]\[z = \frac{9y - 5}{7}\].

Step 6: Interpret the Results

Since there is no direct solution for \(x, y, z\) as real numbers and the derived equations from eliminating \(x\) resulted in dependency, the system does not yield a unique solution. Since \(x\) can be represented via others or \(y\) and \(z\) are dependent, it indicates infinitely many solutions.

Key Concepts

Elimination Method

The Elimination Method is an essential technique for solving systems of linear equations. It involves combining equations to systematically eliminate one of the variables, simplifying the problem to fewer equations. This approach breaks a complex problem down into something more manageable.
The key steps in the elimination method are:

• Choose a variable to eliminate from one or more equation pairs.
• Multiply each equation by suitable coefficients (if necessary) to align the variable terms for easy elimination.
• Add or subtract the equations to remove the chosen variable.
• Solve the resulting simpler equations for the remaining variables.

In this exercise, we eliminated the variable \(x\) by aligning the coefficients of \(x\) in two pairs of equations and subtracted them. This technique simplified the problem to just one equation in terms of \(y\) and \(z\), making it easier to find relationships between these variables.

Infinitely Many Solutions

A system of equations with infinitely many solutions occurs when the equations describe the same geometric object, such as the same line or, in higher dimensions, the same plane.
In practice, it means that there isn't just one specific solution, but rather a whole series of solutions.This usually happens in two common scenarios:

• After eliminating the same variable across various equations, the resulting equations look identical. This indicates they essentially represent the same relationship.
• When the transformation of one equation to align with another through multiplication results in exact equality, except feasible expressions of variables.

In our example, we found that eliminating \(x\) from different pairs of equations resulted in two identical equations, \(9y - 7z = 5\). This implies an infinite number of solutions where \(y\) and \(z\) follow a specific line together in a plane.

Dependent Equations

Dependent equations in a system show a relationship where one equation doesn't add new information – they are effectively the same equation, just expressed differently.
Such dependency indicates that the equations are multiple expressions of the same constraint.In this case, the newly formed equations after eliminating \(x\) were equivalent \(9y - 7z = 5\), making the system dependent.

• This can be inferred when algebraic manipulation transforms different equations into one another.
• In graph terms, multiple lines or planes coincide, showing that the infinite solutions are essentially the same geometric path.
• Understanding these patterns helps in recognizing that the system doesn't change its dimensional space of solutions.

Recognizing dependent equations is crucial because they imply that some variables can be expressed in terms of others, reinforcing the idea of infinite solutions without unique resolution.