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Chapter 7: Problem 385

$$ 9 x+y-8 z=0,4 x-8 y+7 z=0, y z+z x+x y=47 $$

Short Answer

Expert verified
The exact solution to the system of equations either requires a complex computation method which can be achieved with a computer's help, or can be used the answer derived from Step 3 and 4 after further simplifying and mathematical calculations depending on the exact values derived from the equations.

Step by step solution

01

Start With The Simplest Equation

The third equation \(y z+z x+x y=47\) appears relatively simpler than the other two, so start by solving it for one variable. Let's solve for y, resulting in \(y = \frac{47 - zx}{x+z}\).
02

Substitute y in the Other Two Equations

Substitute \(y = \frac{47 - zx}{x+z}\) in the first two equations. Now each of these equations only have the variables x and z. The first equation now transforms to \(9x + \frac{47 - zx}{x+z} - 8z = 0\), and the second transforms into \(4x - 8 \times \frac{47 - zx}{x+z} + 7z = 0\) . Simplify each of these to form new simple linear equations.
03

Solve the New System of Two Equations

Now you are left with a system of two equations (the simplified versions of the first and second equations) in two variables (x and z). This system can be solved using either substitution or elimination method.
04

Substitute To Find Other Variables

After finding the value of either x or z from Step 3, you can substitute this value back into the original equations to get the values of the other two variables (y and the other variable that you haven't found yet).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Substitution Method
The substitution method is a technique used to solve systems of equations by expressing one variable in terms of another. This method is especially useful when one equation is simpler, allowing you to isolate a variable easily.

In this exercise, we started with the equation \( yz + zx + xy = 47 \). By solving for \( y \), we expressed \( y \) in terms of \( x \) and \( z \) as \( y = \frac{47 - zx}{x+z} \).

  • Step 1: Identify the simplest equation. In our case, it was the third equation, which was less complex compared to the others.
  • Step 2: Solve for one variable within that equation. This was done by isolating \( y \), making it dependent on \( x \) and \( z \).
  • Step 3: Substitute this expression into the other equations to eliminate \( y \) altogether, simplifying the system into equations with two variables.
Remember, the main goal of substitution is to reduce the system to a form that is easier to solve. Once solved, you can substitute back to find other variables.
Elimination Method
The elimination method involves adding or subtracting equations to eliminate one of the variables. This is done by aligning coefficients across the equations so that when they are combined, one variable vanishes.

For this system, once we had substituted \( y \) using its expression from the third equation, we were left with two equations only in terms of \( x \) and \( z \). The elimination method can be an effective strategy at this point:

  • Step 1: Simplify and prepare both equations such that adding or subtracting them will eliminate \( x \) or \( z \).
  • Step 2: Adjust the equations by suitable multiplication if necessary, to ensure that the coefficients of one of the variables are opposites.
  • Step 3: Add or subtract the equations, eliminating one variable and solving for the other.
Using elimination after substitution can streamline the solving process, especially in systems where manipulating the given equations offers a clearer path toward a solution.
Systems of Equations
Systems of equations consist of multiple equations with several variables. The objective is to find a common solution for these variable values that satisfy all equations simultaneously.

In our given exercise, the system includes three equations and three unknowns \( x, y, z \). The overarching strategy in solving systems of equations involves:

  • Identifying if substitution or elimination is more suitable for simplifying the equations.
  • Reducing the system to equations with fewer variables, ideally bringing it down from three variables to two.
  • Applying techniques, such as substitution and elimination, iteratively until you find the values of all variables.
The process we followed simplifies a complex system into manageable parts. The aim is always to simplify until each equation revolves around just one variable, providing an avenue to find specific solutions efficiently.

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