Problem 35 - College Algebra and Calculus: An Applied Approach

Question

Solve the system of equations. $$\left\\{\begin{aligned} x &+3 w=4 \\ 2 y-z-w &=0 \\ 3 y-2 w &=1 \\ 2 x-y+4 z &=5 \end{aligned}\right.$$

Step-by-Step Solution

Answer

The solution to the system of equations is \( x = 1, y = 7/6, z = 4/3, w = 1 \)

Step 1: Solve the first equation for x

By manipulating the first equation, it can be rewritten in terms of \( x \). This gives: \( x = 4 - 3w \)

Step 2: Substitute x into the fourth equation

Substitute \( x = 4 - 3w \) into the fourth equation, this will help to simplify the equation system: \( 2(4 - 3w) - y + 4z = 5 \). Simplify this equation to get: \( 8 - 6w - y + 4z = 5 \), or equivalently, \( -6w - y + 4z = -3 \).

Step 3: Combine the second and the new fourth equation

We can add the second equation \( 2y - z - w = 0 \) and the new fourth equation \( -6w - y + 4z = -3 \). The result is: \( -7w + 3z = -3 \)

Step 4: Solve the new equation for z

By manipulating the new equation, it can be rewritten in terms of \( z \). This gives: \( z = (7w - 3)/3 \)

Step 5: Substitute z into the second equation

Substitute \( z = (7w - 3)/3 \) into the second equation. This will give a new equation in terms of only w and y. The new equation is: \( 2y - (7w - 3)/3 - w = 0 \). Simplify it to get: \( 2y - 10w/3 + 1 = 0 \), or equivalently, \( y = (10w - 3)/6 \)

Step 6: Substitute this y into the third equation

Substitute \( y = (10w - 3)/6 \) into the third equation: \( 3(10w - 3)/6 - 2w = 1 \) . Simplify this equation to get the value of \( w \), which is \( w = 1 \)

Step 7: Back substitute w into the y, z, x equations

Back substitute \( w = 1 \) into the equations for \( y, z, x \) to find their values as well. Simplifying gives: \( y = (10*1 - 3)/6 = 7/6, z = (7*1 - 3)/3 = 4/3, x = 4 - 3*1 = 1 \)

Key Concepts

Linear Equations

Linear equations form the building blocks of systems of equations. They are algebraic equations where each term is either a constant or the product of a constant and a single variable. The general form of a linear equation in one variable is expressed as:
\[ ax + b = 0 \]where \(a\) and \(b\) are constants and \(x\) is the variable.
In our exercise, we have multiple such equations, meaning we need to find the solution for several variables.

Some characteristics of linear equations include:

• They graph as straight lines when plotted on a coordinate plane.
• Each equation has a degree of 1, meaning the highest power of the variable is 1.
• They can represent various real-world scenarios, like calculating costs or converting temperatures.

Understanding linear equations is essential because they appear frequently in both simple mathematics and complex real-world problems. They serve as a foundation upon which systems of equations are constructed.

Substitution Method

The substitution method is a crucial concept used to solve systems of linear equations. This technique involves solving one of the equations for one variable and then substituting that expression into the other equations.
This process simplifies the system and reduces the number of variables, making it easier to solve.

Here's how you can execute the substitution method effectively:

• Choose one equation from the system and solve for one variable. For example, solve \(x + 3w = 4\) to get \(x = 4 - 3w\).
• Substitute this expression into the other equations. Replace \(x\) in other equations with \(4 - 3w\).
• Simplify the system using the new equations, and continue substituting until all variables are solved.

This method is especially beneficial when one of the equations is simple and easily manipulated. It allows step-by-step reduction of complexity, making complicated systems manageable.

Simultaneous Equations

Simultaneous equations refer to a set of equations that are solved together since they share common variables. The objective is to find values for the variables that satisfy all the equations in the system simultaneously.
In our exercise, we deal with equations like:

• \(x + 3w = 4\)
• \(2y - z - w = 0\)
• \(3y - 2w = 1\)
• \(2x - y + 4z = 5\)

To solve simultaneous equations, methods like substitution or elimination are often employed:

• Elimination Method: Involves adding or subtracting equations to eliminate a variable, simplifying the process of finding solutions.
• Graphical Method: Entails graphing each equation and finding the intersection point that represents the solution.

Handling simultaneous equations requires understanding their interconnected nature and applying strategies that cater to their unique characteristics.