Problem 17 - Algebra A Combined Function

Question

Solve each system. $$ \left\\{\begin{array}{r} x+2 y-z=5 \\ 6 x+y+z=7 \\ 2 x+4 y-2 z=5 \end{array}\right. $$

Step-by-Step Solution

Answer

The system has infinitely many solutions: \( x = t, y = \frac{12 - 7t}{3}, z = \frac{9 - 11t}{3} \), where \( t \) is any real number.

Step 1: Choose Two Equations to Eliminate a Variable

We start by taking equations from the system. Let's choose the first and third equations:\[ x + 2y - z = 5 \]\[ 2x + 4y - 2z = 5 \]We notice that the third equation is actually a multiple of the first equation, except for the constant term.

Step 2: Recognize Dependent Equations

Since the third equation can be simplified and is a multiple of the first equation, we identify these equations as dependent. This informs us that the system might have infinitely many solutions depending on another variable.

Step 3: Eliminate a Variable from Two Remaining Equations

Let's eliminate \(z\) from the first two equations by adding them:\[ (x + 2y - z) + (6x + y + z) = 5 + 7 \] Which simplifies to:\[ 7x + 3y = 12 \]

Step 4: Solve for One Variable in Terms of Another

Rearrange the equation from Step 3 to solve for \( y \) in terms of \( x \):\[ 3y = 12 - 7x \]\[ y = \frac{12 - 7x}{3} \]

Step 5: Substitute Expression into Original Equation

Substitute \( y = \frac{12 - 7x}{3} \) back into the second original equation \( 6x + y + z = 7 \):\[ 6x + \frac{12 - 7x}{3} + z = 7 \]

Step 6: Solve for Remaining Variables

Multiply the entire equation by 3 to eliminate the fraction:\[ 18x + 12 - 7x + 3z = 21 \]Simplifying, we get:\[ 11x + 3z = 9 \]Now solve for \( z \) in terms of \( x \):\[ 3z = 9 - 11x \]\[ z = \frac{9 - 11x}{3} \]

Step 7: Express System Solution

We express the solution set using a parameter, \( x = t \):\[ x = t, \quad y = \frac{12 - 7t}{3}, \quad z = \frac{9 - 11t}{3} \]Here, \( t \) is any real number, implying an infinite set of solutions.

Key Concepts

Dependent Equations

In a system of equations, dependent equations occur when one equation can be expressed as a multiple of another. This connection implies that they essentially represent the same line or plane when graphed. In the exercise, the first and third equations were dependent:

• The first equation: \(x + 2y - z = 5\)
• The third equation: \(2x + 4y - 2z = 5\)

Notice that the third equation is a simple multiple of the first one. This means they do not add any new information to the system. Understanding dependency helps in identifying whether a system might have infinite solutions, as these equations don't provide unique intersections.

Infinitely Many Solutions

When a system of equations has infinitely many solutions, it means the equations describe the same geometric entity, like overlapping planes or lines. In practical terms:

• The system doesn't intersect at a single point.
• Instead, a "line of solutions" or "plane of solutions" is formed.

In the context of the problem, since the equations are dependent, solving the system shows a parameter can be introduced, leading to infinitely many solutions. The presence of dependent equations like \(x + 2y - z = 5\) and its multiple \(2x + 4y - 2z = 5\) indicates overlap.

Elimination Method

The elimination method is a technique used to remove one variable from a system of equations. This is achieved by adding or subtracting equations to cancel out a particular variable. Here’s how it works:1. Choose two equations that make it easy to eliminate a variable through addition or subtraction.2. Manipulate the equations to align terms.3. Add or subtract the equations to "eliminate" one variable, simplifying the system.In the exercise, we first eliminated \(z\) by adding the first two equations:\[(x + 2y - z) + (6x + y + z) = 5 + 7\]This resulted in a simpler equation \(7x + 3y = 12\), which no longer included \(z\), making further calculations manageable.

Parametric Solutions

A parametric solution expresses the variables of a system in terms of a single parameter, often denoted as \(t\). This parameter spans a range, offering a way to express infinitely many solutions. For the problem at hand:

• We chose \(x = t\) where \(t\) can be any real number.
• This choice provided solutions for \(y\) and \(z\). Specifically:
  • \(y = \frac{12 - 7t}{3}\)
  • \(z = \frac{9 - 11t}{3}\)

Parametric solutions are powerful in expressing the complete set of solutions concisely. They make it easy to understand how solutions vary with \(t\), showing the interdependence among variables in the system.