Problem 2 - Algebra and Trigonometry

Question

\(1-2\) . These exercises refer to the following system. $$ \left\\{\begin{aligned} x-y+z &=2 \\\\-x+2 y+z &=-3 \\ 3 x+y-2 z &=2 \end{aligned}\right. $$ To eliminate \(x\) from the third equation, we add________ times the first equation to the third equation. The third equation becomes______=____

Step-by-Step Solution

Answer

Add \(-3\) times, the third equation becomes \(4y - 5z = -4\).

Step 1: Identify the Target Equation and Required Transformation

To eliminate the variable \(x\) from the third equation, we begin by examining both the first and third equations. We specifically look for a way to combine these equations such that \(x\) is eliminated from the third equation.

Step 2: Compare and Match Coefficients

The third equation is \(3x + y - 2z = 2\). To eliminate \(x\), we need to add or subtract multiples of the first equation. The first equation is \(x - y + z = 2\). We want the coefficient of \(x\) in the third equation to become zero after adding or subtracting a multiple of the first equation.

Step 3: Calculate the Necessary Multiple

Since the third equation has \(3x\) and the goal is to cancel \(x\) from the first equation \(x - y + z\), we need \(3(\text{some multiple}) + 1x = 0\). Thus, we should use \(-3\) times the first equation, resulting in \(-3x + 3y - 3z = -6\).

Step 4: Perform the Addition/Subtraction

Add \(-3\) times the first equation to the third equation:\[3x + y - 2z + (-3x + 3y - 3z) = 2 + (-6)\]This simplifies to:\[(3x - 3x) + (y + 3y) + (-2z - 3z) = 2 - 6\]

Step 5: Write the New Third Equation

After carrying out the addition, the terms with \(x\) cancel each other, yielding:\[4y - 5z = -4\]Thus, the third equation becomes \(4y - 5z = -4\).

Key Concepts

Elimination Method

The elimination method is a technique used to solve systems of linear equations. It involves manipulating the equations to systematically eliminate one variable at a time.
This makes the system easier to solve. In our exercise, we wanted to eliminate the variable \(x\) from an equation by using other equations in the system.Using the elimination method involves several steps. First, identify which variable you wish to eliminate. In our case, it was \(x\) from the third equation. Next, adjust and combine the equations so that the coefficient of the variable you are trying to eliminate is matched and inverted in the other equation. Here, we added \(-3\) times the first equation to the third.

• Choose a variable to eliminate.
• Manipulate one equation to offset the selected variable's coefficient.
• Add or subtract equations to eliminate the chosen variable.

Effective use of the elimination method can greatly simplify solving systems of equations, allowing easier solving of the remaining equations after one variable is removed. This is especially useful in larger systems with more than two variables.

Linear Equations

Linear equations are equations involving variables with no exponents greater than one. In a system of linear equations, each equation represents a straight line when graphed on a coordinate plane. A common method of solving linear equations is to consider them as intersecting lines in a multi-dimensional space.In our problem, we deal with a system of three linear equations:- \(x - y + z = 2\) - \(-x + 2y + z = -3\) - \(3x + y - 2z = 2\)Each of these equations represents a plane in three-dimensional space. Solving the system means finding the points where all planes intersect. By using techniques like the elimination method, we numerically find those points without needing graphical representations.
Understanding linear equations helps grasp larger systems, as any solution must satisfy all constituent equations simultaneously.

Algebra

Algebra is a branch of mathematics dealing with symbols and the rules for manipulating these symbols. It provides a powerful set of tools and techniques for solving equations, such as the systems discussed in our exercise.In this system of three equations, algebra helps us perform operations such as addition, subtraction, and multiplication on these symbolic equations to reduce them. Applying algebraic rules and operations is key to re-writing equations in simpler forms, like transforming \(3x = -3(x - y + z)\) into \(-3x + 3y - 3z\), allowing us to eliminate \(x\) successfully.By using algebra, we can logically deduce solutions by:

• Identifying variables and constants.
• Applying arithmetic operations among equations.
• Rearranging and simplifying the expressions.

Algebra makes these processes systematic and understandable, letting us handle complex systems of equations efficiently and accurately.