Problem 6 - Algebra A Combined Function

Question

Solve each system. $$ \left\\{\begin{array}{r} 2 x-3 y+z=5 \\ x+y+z=0 \\ 4 x+2 y+4 z=4 \end{array}\right. $$

Step-by-Step Solution

Answer

The solution is \((-3, -2, 5)\).

Step 1: Simplify the third equation

First, notice that the third equation \(4x + 2y + 4z = 4\) can be simplified by dividing all terms by 2, as they have a common factor. It simplifies to: \(2x + y + 2z = 2\). This gives a simpler system to work with.

Step 2: Set up the modified system

Now the system is: \[\begin{cases} 2x - 3y + z = 5 \ x + y + z = 0 \ 2x + y + 2z = 2 \end{cases} \] Here, all equations are simplified and ready for elimination.

Step 3: Eliminate variable from two equations

Subtract the second equation \(x + y + z = 0\) from the first equation \(2x - 3y + z = 5\) to eliminate \(z\). 1. \((2x - 3y + z) - (x + y + z) = 5 - 0\) 2. This results in \(x - 4y = 5\). Label this Equation (4).

Step 4: Eliminate the same variable from another pair of equations

Subtract the second equation \(x + y + z = 0\) from the third equation \(2x + y + 2z = 2\) to also eliminate \(z\) from here. 1. \((2x + y + 2z) - (x + y + z) = 2 - 0\) 2. This results in \(x + z = 2\). Label this Equation (5).

Step 5: Solve the reduced system of equations

Now, solve the equations obtained: \[ \begin{cases} x - 4y = 5 \ x + z = 2 \end{cases} \] 1. From Equation (5), express \(x\) in terms of \(z\): \(x = 2 - z\). 2. Substitute \(x = 2 - z\) into Equation (4): \((2 - z) - 4y = 5\) \(2 - z = 5 + 4y\) \(z = 2 - 5 - 4y\) \(z = -3 - 4y\).

Step 6: Find values for each variable

From the previous step, substitute \(z = -3 - 4y\) back into \(x = 2 - z\) to find \(x\): 1. \(x = 2 - (-3 - 4y) = 2 + 3 + 4y\) 2. \(x = 5 + 4y\). Put \(x = 5 + 4y\) and \(z = -3 - 4y\) into \(x + y + z = 0\): 1. \((5 + 4y) + y + (-3 - 4y) = 0\) 2. Solve for \(y\): \(5 + y - 3 = 0 \rightarrow y = -2\). Finally, substitute \(y = -2\) back to get \(x\) and \(z\): - \(x = 5 + 4(-2) = -3\) - \(z = -3 - 4(-2) = 5\).

Step 7: Verify the solution

Double-check the solution \((-3, -2, 5)\) by substituting these values back into the original equations to confirm they satisfy all equations. 1. Check first equation: \(2(-3) - 3(-2) + 5 = -6 + 6 + 5 = 5\) which is true. 2. Check second equation: \(-3 + (-2) + 5 = 0\), also true. 3. Check third equation (simplified): \(2(-3) + (-2) + 2(5) = -6 - 2 + 10 = 2\), correct. Since all hold true, the solution is verified.

Key Concepts

Elimination Method

The elimination method is a strategy used to solve a system of linear equations. The idea is to eliminate one of the variables by combining the equations in such a way that it disappears. This allows us to work with a smaller system of equations that is easier to solve.

Let's break down the steps:

• Identify pairs of equations in which a variable can be easily eliminated by adding or subtracting them.
• Perform the addition or subtraction operation to remove one variable.
• Solve the resulting simpler system.

The elimination method is beneficial because it systematically reduces the complexity of the system, making it easier to find the values of the remaining variables. It's a straightforward technique once you understand how to manipulate the equations to eliminate unwanted variables.

Linear Equations

Linear equations are mathematical expressions that model relationships between two or more variables with constant rates of change. In their simplest form, they take the shape of a straight line when graphed on a coordinate plane.

Key characteristics of linear equations include:

• The general form of a linear equation is \( ax + by = c \), where \( a, b, \) and \( c \) are constants.
• Solutions for these equations represent points where the line crosses the axes or other lines.
• They describe real-world situations, such as calculating distance, speed, or cost predictions based on given rates.

Understanding linear equations is crucial as they form the basis for solving more complex systems like the one in the original exercise. Recognizing their structure can simplify identifying solutions through methods such as substitution or elimination.

System of Equations

A system of equations consists of two or more equations that share common variables. The goal is to find values for these variables that satisfy all equations in the system simultaneously.

Features of a system of equations include:

• Simplifying equations by identifying common terms or factors, which make handling overall systems easier.
• Using methods like substitution, elimination, or graphical interpretation to solve them.
• Providing solutions that represent points of intersection for the equations when graphed.

Understanding systems of equations is essential as they are applied in different contexts, such as engineering, economics, and physics. Solving these systems can reveal insights about multiple interacting components within real-world problems, allowing for deeper comprehension and prediction capabilities.