Problem 14 - Algebra and Trigonometry

Question

Perform an operation on the given system that eliminates the indicated variable. Write the new equivalent system. $$ \left\\{\begin{aligned} x-4 y+z &=3 \\ y-3 z &=10 \\ 3 y-8 z &=24 \end{aligned}\right. $$ Eliminate the \(y\) -term from the third equation.

Step-by-Step Solution

Answer

The new system is: 1. \( x - 4y + z = 3 \) 2. \( y - 3z = 10 \) 3. \( z = -6 \).

Step 1: Write the Original System of Equations

The initial system of equations is: 1. \( x - 4y + z = 3 \) 2. \( y - 3z = 10 \) 3. \( 3y - 8z = 24 \). We are tasked with eliminating the \( y \)-term from the third equation.

Step 2: Solve for \( y \) in terms of \( z \) from Equation 2

From equation 2, \( y - 3z = 10 \). Solve for \( y \): \[ y = 3z + 10 \] This expression will be used to eliminate \( y \) in equation 3.

Step 3: Substitute \( y = 3z + 10 \) into Equation 3

Substitute the expression for \( y \) from Step 2 into Equation 3: \[ 3(3z + 10) - 8z = 24 \]. Expand and simplify the expression to remove \( y \) from the equation.

Step 4: Simplify the New Equation

Simplify the equation from step 3: \[ 9z + 30 - 8z = 24 \] Combine like terms: \[ z + 30 = 24 \] Solve for \( z \) by subtracting 30 from both sides: \[ z = 24 - 30 = -6 \].

Step 5: Write the New System

Now that we have eliminated \( y \) from equation 3, let's write the new equivalent system: 1. \( x - 4y + z = 3 \) 2. \( y - 3z = 10 \) 3. \( z = -6 \). This new system does not contain \( y \) in the third equation.

Key Concepts

System of Equations

Understanding the concept of a **system of equations** is crucial when dealing with multiple mathematical statements. This system of equations involves three different linear equations:

• Equation 1: \( x - 4y + z = 3 \)
• Equation 2: \( y - 3z = 10 \)
• Equation 3: \( 3y - 8z = 24 \)

The objective is to find the values of the variables \( x \), \( y \), and \( z \) that satisfy all equations simultaneously. It's like solving a puzzle where each equation is a piece, and together they reveal the complete picture. Typically, systems of equations can be solved using different methods such as substitution, elimination, or graphing. Each method provides a different approach to find the common solution for the variables involved. In this exercise, the focus is on eliminating the variable \( y \) from one of the equations to simplify the problem.

Variable Elimination

**Variable elimination** is a widely used technique in which we aim to simplify a system by reducing the number of variables in any given equation. It can make finding solutions easier by focusing on fewer variables at a time. In the provided exercise, we are tasked with eliminating the variable \( y \) from the third equation, \( 3y - 8z = 24 \). Here’s how it’s done:

• First, solve for the variable you wish to eliminate in terms of other variables in the system. In this case, we solve Equation 2: \( y - 3z = 10 \) to get \( y = 3z + 10 \).
• Next, substitute this expression for \( y \) into the equation you are targeting, which is Equation 3.
• Finally, simplify the equation to effectively eliminate \( y \) and reduce the system to two relevant variables.

Through this process, you've effectively reduced the complexity of the problem by removing a variable from one of the equations. Variable elimination helps in isolating unknowns, making it easier to solve complex problems incrementally.

Equation Simplification

**Equation simplification** is a process undertaken to simplify mathematical equations into a more manageable form. It's especially useful for making equations easier to solve and understand, by clearer expression of key components. Here’s how the simplification process worked in solving the given system:

• After substituting \( y = 3z + 10 \) into the third equation, we have: \( 3(3z + 10) - 8z = 24 \).
• Expanding this expression involves multiplying through: \( 9z + 30 - 8z \).
• Combine like terms to further simplify: \( z + 30 = 24 \).
• Finally, solve for \( z \) by isolating it on one side of the equation: subtract 30 from both sides to get \( z = -6 \).

By reducing the equation to a simpler form, we directly found the value of one variable, \( z = -6 \). This simplification step is key in breaking down complex systems of equations and arriving at solutions more efficiently. It highlights the inherent power of breaking down mathematical operations to uncover straightforward answers.