Problem 45 - Algebra A Combined Function

Question

Multiply both sides of equation (1) by -3 , and add the resulting equation to equation (2). $$ \begin{array}{r} x+2 y-z=0 \\ 3 x+y-z=2 \end{array} $$

Step-by-Step Solution

Answer

The resulting equation is \(-5y + 2z = 2\).

Step 1: Identify Equation (1)

The first equation is given as \( x + 2y - z = 0 \). This will be used to manipulate and combine with the second equation.

Step 2: Multiply Equation (1) by -3

Multiply every term in the first equation by \(-3\) to manipulate the original equation. This gives: \(-3(x + 2y - z) = -3 \times 0\). Simplifying, we get \(-3x - 6y + 3z = 0\).

Step 3: Write Equation (2)

The second equation is given as \(3x + y - z = 2\). This will be added to the transformed equation from Step 2.

Step 4: Add Transformed Equation (1) to Equation (2)

Combine equation from Step 2 with Equation (2): \((-3x - 6y + 3z) + (3x + y - z) = 0 + 2\). Simplifying the left side, \(-3x + 3x = 0\), \(-6y + y = -5y\), and \(3z - z = 2z\). This simplifies to \(-5y + 2z = 2\).

Step 5: Simplify the Equation

The resulting simplified equation from combining the two is \(-5y + 2z = 2\). This shows the relation between variables \(y\) and \(z\).

Key Concepts

Equation Manipulation

Equation manipulation involves performing operations to change the form of an equation while keeping its solutions the same. In the exercise, we tackle this by multiplying terms of an equation by a constant. Here’s why it’s important:
Every operation should maintain the balance of the equation, just like balancing a scale. We multiplied Equation (1) by \(-3\) to align terms with Equation (2), allowing for easier combination.
Such techniques are fundamental in isolating or eliminating variables in a system of equations. This method opens avenues for solving the equations more efficiently in subsequent steps.

Systems of Equations

A system of equations is a collection of two or more equations with the same set of unknowns. The main goal when dealing with systems is to find values for these unknowns that satisfy all equations simultaneously.
In our exercise, the system comprises the equations: \(x + 2y - z = 0\) and \(3x + y - z = 2\). These two equations form a linear system, and our task was to manipulate and combine them to find relationships between the variables. Solving systems often involves finding the point where the equations intersect, which represents the solution to the system.
This concept is critical in understanding more complex scenarios in mathematics, from vectors in physics to economic models in social sciences.

Equation Simplification

Simplifying equations is about reducing equations into simpler forms without changing their solutions. The idea is to make them easier to solve or understand.
In the step-by-step solution, \(-3x - 6y + 3z = 0\) was added to \(3x + y - z = 2\) and simplified by combining like terms. The process led to the simpler form \(-5y + 2z = 2\).
This simplification process involves:

• Recognizing like terms, such as \(x\) terms or \(y\) terms.
• Combining these terms to reduce complexity.

This is crucial because it reduces potential errors in solving and aids in better visualization and understanding of the relationships between variables.

Algebraic Methods

In algebra, various methods help us to solve equations and systems of equations, ranging from substitution to elimination.
The exercise employs elimination as a strategy where one variable is removed by adding or subtracting equations, as evident when \(x\) was eliminated in our problem. It transformed a two-variable problem into one that is simpler to handle.
Algebraic methods are powerful as they provide precise, methodical ways to handle complex equations and shed light on underlying mathematical relationships.