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Chapter 8: Problem 56

Write a linear equation in three variables that is satisfied by all three of the given ordered triples. $$(0,0,2),(0,1,0),(1,0,0)$$

Short Answer

Expert verified
The linear equation is: \(2x + 2y + z = 2\).

Step by step solution

01

Identify the general form of a linear equation in three variables

The general form of a linear equation in three variables is: \[ ax + by + cz = d \]where \( a, b, \) and \( c \) are the coefficients and \( d \) is the constant term.
02

Substitute the first ordered triple into the equation

Substitute the coordinates of the first ordered triple \(0, 0, 2\) into the equation:\[ a(0) + b(0) + c(2) = d \]which simplifies to:\[ 2c = d \]
03

Substitute the second ordered triple into the equation

Substitute the coordinates of the second ordered triple \(0, 1, 0\) into the equation:\[ a(0) + b(1) + c(0) = d \]which simplifies to:\[ b = d \]
04

Substitute the third ordered triple into the equation

Substitute the coordinates of the third ordered triple \(1, 0, 0\) into the equation:\[ a(1) + b(0) + c(0) = d \]which simplifies to:\[ a = d \]
05

Find the relationship between coefficients and constant term

From the given substitutions, we have:\[ 2c = d \]\[ b = d \]\[ a = d \]This means \( d = a = b = 2c \). Let us use \( a \), \( b \) and \( c \) in terms of \( d \). We can take \( d = k \) for some constant \( k \). So, we have:\[ a = b = k \text{ and } c = \frac{k}{2} \]
06

Write the final equation

Substitute the coefficients back into the general form:\[ kx + ky + \frac{k}{2}z = k \]Divide through by \( k \):\[ x + y + \frac{1}{2}z = 1 \]Multiplying every term by 2 for simplicity:\[ 2x + 2y + z = 2 \]

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Key Concepts

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

Ordered Triples
When dealing with linear equations in three variables, ordered triples play a crucial role. An ordered triple consists of three numbers written in a specific order, usually denoted as \( (x, y, z) \). Each number represents the value of a corresponding variable in the equation. In the given problem, the ordered triples are \( (0,0,2) \), \( (0,1,0) \), and \( (1,0,0) \).
Ordered triples act as solutions to the linear equation. Each triple should satisfy the equation when the values are substituted for \( x \), \( y \), and \( z \).
Understanding this helps us check whether a proposed equation is correct. If each ordered triple satisfies the equation, then our equation is correct for those specific points.
Coefficients
In a linear equation, coefficients are the numbers placed in front of the variables. For example, in the equation \[ ax + by + cz = d \], \( a \), \( b \), and \( c \) are the coefficients. These coefficients determine how much each variable contributes to the value of the equation.
In the given problem, the values we identified for \( a, b, \) and \( c \) are critical:
  • First, we found that \[2c = d\]
  • Then, \[b = d\], and
  • Finally, \[a = d\].
This established a relationship among the coefficients and the constant term, letting us express them in terms of a single variable, \( k \). We used these to rewrite the equation systematically.
Standard Form
The standard form of a linear equation in three variables is \[ ax + by + cz = d \]. This form makes it easy to understand how each variable and its coefficient relate to the constant term \( d \).
In the given problem, we started with this general form and substituted the given ordered triples to derive the relationships between the coefficients and the constant term. Through the process, we simplified the initial \[ ax + by + cz = d \] to a more manageable \[ 2x + 2y + z = 2 \]. This simpler form makes it easier to check whether any point \( (x, y, z) \) would satisfy the equation.
Understanding the standard form is crucial as it allows you to convert between different forms of equations and easily check solutions.

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