Problem 41 - Algebra and Trigonometry

Question

35–46 Solve the system of linear equations. $$\left\\{\begin{array}{rrr}{x+y+2 z-} {w=} {-2} \\ {3 y+z+2 w} {=} {2} \\\ {x+y} \qquad {+3 w=} {2} \\ {-3 x} \qquad {+z+2 w=} {5}\end{array}\right.$$

Step-by-Step Solution

Answer

The system's solution requires iterative reconciliation of variable dependencies.

Step 1: Reorganize Equations

Let's rewrite each equation for clarity:1. \( x + y + 2z - w = -2 \)2. \( 3y + z + 2w = 2 \)3. \( x + y + 3w = 2 \)4. \( -3x + z + 2w = 5 \).

Step 2: Express Variables (Step 1)

From the third equation, \( x + y + 3w = 2 \), we can express one of the variables in terms of the others. Let's solve for \( x \):\[ x = 2 - y - 3w \].

Step 3: Substitute to Eliminate x (Step 2)

Substitute \( x = 2 - y - 3w \) into the first equation:\[ (2 - y - 3w) + y + 2z - w = -2 \].Simplify to get:\[ 2 + 2z - w - 3w = -2 \].\[ 2 + 2z - 4w = -2 \].\[ 2z - 4w = -4 \].Divide through by 2:\[ z - 2w = -2 \] (we'll call this equation 5).

Step 4: Substitute to Eliminate x (Step 3)

Substitute \( x = 2 - y - 3w \) into the fourth equation:\[ -3(2 - y - 3w) + z + 2w = 5 \].Simplify:\[ -6 + 3y + 9w + z + 2w = 5 \].\[ 3y + 11w + z = 11 \] (we'll call this equation 6).

Step 5: Simplify Using Two Substituted Equations

We now have:1. \( z - 2w = -2 \)2. \( 3y + z + 11w = 11 \).Substitute \( z = -2 + 2w \) from the first equation into the second:\[ 3y + (-2 + 2w) + 11w = 11 \].Simplify:\[ 3y - 2 + 2w + 11w = 11 \].\[ 3y + 13w = 13 \].Divide by 3:\[ y = rac{13 - 13w}{3} \].

Step 6: Substitute y Back to Find z

Substitute \( y = \frac{13 - 13w}{3} \) back into \( z - 2w = -2 \):Recall from earlier simplification:\[ z = 2w - 2 \].To find \( w \), let's substitute into any convenient equation, potentially revisiting full substitutions to maintain equation consistency.

Step 7: Solve for Remaining Variables

To proceed, match values:Reconsider: from \( 3y + z + 11w = 11 \):Plugged expressions will now help iteratively deduce equilibrium.Track initial veins again: if \( z = 2w - 2 \), now from base definitions.

Key Concepts

System of Equations

A system of equations is a collection of two or more equations with the same set of variables. In our exercise, we have four equations with variables: \( x, y, z, \) and \( w \). The goal is to find values for these variables that make all the equations true simultaneously. Solving systems of linear equations is a fundamental concept in algebra, often used to determine the point of intersection of lines or planes. There are several methods to solve these systems, including substitution and elimination, which we'll explore in the following sections.
Understanding the system of equations is key to approaching problems involving multiple equations and variables.

• Identify all the variables and equations in the system.
• Determine a strategy (like substitution or elimination) based on the structure of the equations.
• Work step-by-step, focusing on simplifying the system to find solutions efficiently.

Substitution Method

The substitution method involves solving one of the equations for one variable and then substituting that expression into the other equations. This reduces the number of variables and equations gradually. For example, from our step-by-step solution:

• We first solve for \( x \) in the equation \( x + y + 3w = 2 \), resulting in \( x = 2 - y - 3w \).
• This expression for \( x \) can then be substituted into other equations to eliminate \( x \) and simplify the system.

This method is particularly useful when one of the equations is easily solvable for a variable, simplifying manipulation and reducing complexity. It requires patience as each substitution leads to simpler systems, helping uncover the values of other variables step-by-step.

Elimination Method

The elimination method, also known as the addition method, involves adding or subtracting equations to eliminate one of the variables. This helps in focusing on fewer variables, making it easier to solve the system. For instance, after expressing \( x \) in terms of the other variables, we handled:

• The simplified equation \( 2z - 4w = -4 \), derived by eliminating \( x \) through substitution. This is equivalent to adding equations with the variable \( x \) removed.

By manipulating these new equations, such as combining or multiplying equations to cancel out certain variables, the system is further simplified. Once a variable is eliminated, it aids in solving for the remaining variables straightforwardly. The key is to maintain a clear understanding of the relationships between the equations.

Algebraic Manipulation

Algebraic manipulation involves rearranging and simplifying equations to make them easier to solve. It includes operations such as combining like terms, distributing, and factoring. In our solution:

• The initial reshaping of equations like \( 3y + z + 11w = 11 \), aligns the variables neatly.
• Manipulating \( z - 2w = -2 \) to \( z = 2w - 2 \) simplifies further substitutions.

Such manipulation is integral for ensuring equations are in a workable form. Proper manipulation involves carefully tracking each algebraic step to avoid common pitfalls like arithmetic mistakes or loss of variable terms. It forms the backbone of solving any algebraic problem, allowing focus on solving one variable at a time.