Problem 20 - Algebra A Combined Function

Question

Solve each system. $$ \left\\{\begin{array}{r} 4 x+y-z=8 \\ x-y+2 z=3 \\ 3 x-y+z=6 \end{array}\right. $$

Step-by-Step Solution

Answer

\(x = 1\), \(y = \frac{3}{2}\), \(z = 1\)

Step 1: Choose a Variable to Eliminate

We'll start by eliminating the variable \(y\) from the equations. This will simplify our system. Begin by focusing on the first and second equations.

Step 2: Eliminate y from First and Second Equations

Multiply the second equation by 1 to keep it the same:\[ x - y + 2z = 3 \]Then subtract it from the first equation:\[ (4x + y - z) - (x - y + 2z) = 8 - 3 \]Simplifying, we get:\[ 3x + 2y - 3z = 5 \]Eliminate \(y\) from the first and third equations next.

Step 3: Eliminate y from First and Third Equations

To eliminate \(y\) from the first and third equations:Multiply the third equation by 1 to keep it the same:\[ 3x - y + z = 6 \]Subtract it from the first equation:\[ (4x + y - z) - (3x - y + z) = 8 - 6 \]Simplifying, we get:\[ x + 2y - 2z = 2 \]

Step 4: Solve the New System

Now, solve the new system of equations obtained in Step 2 and Step 3:1. \(3x + 2z = 5\)2. \(x + 2y - 2z = 2\)Use these two equations to find values for \(x\) and \(z\).

Step 5: Solve for x and z

From the equation \(3x + 2z = 5\), express \(z\) in terms of \(x\):1. \(2z = 5 - 3x\)2. \(z = \frac{5 - 3x}{2}\)Substitute \(z = \frac{5 - 3x}{2}\) into the second equation \(x + 2y - 2z = 2\) and solve for \(y\).

Step 6: Substitute and Solve

Substitute \(z = \frac{5 - 3x}{2}\) into \(x + 2y - 2z = 2\):\[ x + 2y - 2(\frac{5 - 3x}{2}) = 2 \]Simplify:\[ x + 2y - (5 - 3x) = 2 \]\[ x + 2y - 5 + 3x = 2 \]\[ 4x + 2y = 7 \]Now solve for \(y\).

Step 7: Solve for y

Divide the equation \(4x + 2y = 7\) by 2 to simplify:\[ 2x + y = \frac{7}{2} \]Solving for \(y\):\[ y = \frac{7}{2} - 2x \]Now, substitute this value back into the expression for \(z\) to find the actual values of \(x\), \(y\), and \(z\).

Step 8: Solve for x, y, and z

Substitute back the values: if \(x = 1\):\[ y = \frac{7}{2} - 2(1) = \frac{7}{2} - 2 = \frac{3}{2} \]\[ z = \frac{5 - 3(1)}{2} = \frac{5 - 3}{2} = 1 \]So that means \(x = 1\), \(y = \frac{3}{2}\), and \(z = 1\). These values satisfy the conditions.

Key Concepts

Elimination Method

The elimination method is a powerful technique used to solve systems of equations. It involves manipulating equations in such a way as to "eliminate" one of the variables. This simplifies the system and makes it easier to solve. By strategically adding or subtracting equations, one variable can be removed, leaving a simpler system behind.

In the given exercise, we are tackling a system of three equations:

• \( 4x + y - z = 8 \)
• \( x - y + 2z = 3 \)
• \( 3x - y + z = 6 \)

Here, the goal is to eliminate the variable \( y \). This choice is typically based on simplicity, as \( y \) appears with minimal coefficients. Thus, it's practical to focus on eliminating \( y \) first, simplifying the complexities of aligning coefficients. By methodically working through each step, the equations become more manageable.

Solve for Variables

Once a variable is eliminated, the focus shifts to solving for the remaining variables. Now that we've eliminated \( y \), the system reduces:

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

We can express \( z \) in terms of \( x \) from the first equation. Solving \( 3x + 2z = 5 \) for \( z \) gives us \( z = \frac{5 - 3x}{2} \).

Next, substituting this expression for \( z \) into the second equation \( x + 2y - 2z = 2 \), we solve for \( y \). By simplifying, we get \( 4x + 2y = 7 \), which further simplifies to \( y = \frac{7}{2} - 2x \).

These manipulations bring us to a point where only straightforward algebra is needed, leading us to the precise values for the variables, aligning perfectly with the solution requirements.

Step-by-Step Solution

Breaking down the solution into smaller, manageable steps is crucial to understanding. Here's a structured walkthrough:
**Step 1:** Choose which variable to eliminate. In our problem, we've opted for \( y \). Focusing on equations with \( y \), like the first and second, aligns the coefficients effectively.**Step 2:** Eliminate \( y \) by subtracting one equation from the other. For example, subtract \( (x - y + 2z) \) from \( (4x + y - z) \) to achieve simplification.**Step 3:** Create a new system from these elimination steps, such as \( 3x + 2z = 5 \), that does not include \( y \). Use this to solve for another variable in terms of another one, say \( z \) in terms of \( x \).**Step 4:** Substitute this relationship into other equations to solve systematically for remaining variables like \( y \), using simplification strategies like dividing through or expressing one variable in terms of the others.

• Ensure steps are followed in a logical sequence to avoid errors and keep track of each variable's role during the manipulation.

By meticulously observing, and repeating these organized steps, any complex system of equations can be tackled with confidence and clarity.