Problem 38 - College Algebra

Question

Exercises 37-39 will help you prepare for the material covered in the first section of the next chapter. Solve the system: $$\left\\{\begin{aligned}w-x+2 y-2 z &=-1 \\\x-1 y+z &=1 \\\y-z &=1 \\\z-&-3\end{aligned}\right.$$ Express the solution set in the form \(\\{(\boldsymbol{x}, \boldsymbol{x}, \boldsymbol{y}, \boldsymbol{z})\\} .\) What makes it fairly easy to find the solution?

Step-by-Step Solution

Answer

The solution set is \(\{(-2, 2, -2, -3)\}\)

Step 1: Isolate variable z from the last equation

From the equation \(z = -3\), it is clear that z is isolated. Hence, the value of z is -3.

Step 2: Substitute z in the other equations

Now, substitute \(z = -3\) in the equations \(y - z = 1\) and \(x - y + z = 1\). This will give us \(y = -3 + 1 = -2\) and \(x = -2 - (-3) + 1 = 2\)

Step 3: Substitute x, y, z in the first equation

Substitute \(x = 2\), \(y = -2\), \(z = -3\) into the first equation \(w - x + 2y - 2z = -1\). This will give us \(w = -1 + 2 - 2*(-2) + 2*(-3) = -2\)

Step 4: Express the solution set

After finding the values of x, y, z and w. Solutions can be written in the form \(\{(w, x, y, z)\}\)

Key Concepts

linear equations

Linear equations are mathematical expressions that show a relationship between variables, often portrayed in lines when graphed. In this system, the equations involve variables denoted by letters such as \(w\), \(x\), \(y\), and \(z\). These variables are either added, subtracted, or set equal to a constant value.

This specific set of linear equations shows how multiple algebraic terms are related. Each equation helps us understand how variables correlate in a two-dimensional plane. For example, in the system given, each equation depicts a linear relationship amongst four variables. Applying the operations in each equation aims to balance them to achieve a solution.

Linear equations are fundamental to solving systems, as they allow the determination of unknown values by manipulating expressions. Understanding the basic properties of each equation helps to use techniques like substitution or elimination efficiently. A key once more is recognizing that all expressions are linear, meaning each variable is raised only to the power of one.

variable isolation

Variable isolation is a crucial step in solving systems of equations, where the aim is to express one variable in terms of others, or constants. This makes it easier to simplify the complex relationships into solvable parts.

In the provided solution, the last equation \(z = -3\) is already isolated, making our first step straightforward. By identifying such opportunities early on, we can simplify the work ahead since it makes substitution easier.

To isolate a variable effectively in any equation, you usually follow these steps:

• Identify the variable you want to isolate.
• Perform operations that cancel out other terms surrounding this variable.
• Maintain balance between the two sides of the equation by performing the same operation on both sides.

By isolating one of the variables, you dramatically reduce the complexity of the other equations, paving the way for simpler substitution and final solution.

substitution method

The substitution method involves replacing a variable in one equation with an expression derived from another equation. This method can simplify multiple-variable systems by reducing the number of equations you need to solve consecutively.

In the exercise, once \(z\) was isolated as \(-3\), this value was substituted back into the other equations—\(y - z = 1\) and \(x - y + z = 1\). Here’s a step-by-step:

• Take the isolated variable from one equation and substitute its known value into the others.
• Solve new equations with this substituted value to find the actual values of other variables.
• Continue substituting until all variables are known.

This method is highly effective when one of the equations easily allows for isolation of a variable, streamlining the path to finding all unknown variables without resolving numerous linear equations simultaneously.

solution set

A solution set is the collection of all possible solutions that satisfy the system of equations. Once all variables are determined, they together form the solution to the system.

In this exercise, after using variable isolation and substitution, the calculated values \(w = -2\), \(x = 2\), \(y = -2\), and \(z = -3\) can be expressed together in the form \(\{(w, x, y, z)\}\).

The utility of expressing solutions this way is its clarity—it gives a complete description of the intersection point where all individual equations in the system hold true. A solution set provides a snapshot of the resolution of the entire system, showing that each equation was resolved accurately.