Problem 81 - Precalculus with Limits

Question

Finding Systems of Linear Equations In Exercises \(79 - 82 ,\) find two systems of linear equations that have the ordered triple as a solution. (There are many correct answers.) $$ \left( - 6 , - \frac { 1 } { 2 } , - \frac { 7 } { 4 } \right) $$

Step-by-Step Solution

Answer

The two systems of linear equations that have \((-6, -1/2, -7/4)\) as a solution can be:System 1:1. \(x + y + z = -9\)2. \(2x + 2y + z = -15\)System 2:1. \(-2x -2y -2z = 11\)2. \(3x + 2y + z = -21\)Remember, there are infinite correct solutions for this problem. These are just two examples of such systems.

Step 1: Starting with the first system of linear equations

Let's create a system using the simple forms for linear equations i.e., standard form \( ax + by + cz = d \) where \( a, b, c, d \) are constants. Using the given ordered triple \( x = -6 , y = -1/2 , z = -7/4 \) as solution, we can choose some arbitrary values for \( a, b, c, d \) in such a way that the equations are satisfied. Let's start with \( a = b = c = 1 \) and \( d = -6 -1/2 -7/4 \) for the first equation, and \( a = b = 2, c = 1, d = -12 -1 -7/2 \) for the second one. This gives us the first system of equations:1. \( x + y + z = -6 -1/2 -7/4 = -9 \)2. \( 2x + 2y + z = -12 -1 -7/2 = -15 \)

Step 2: Creating the second system of linear equations

Now let's create a second system of equations, again using the standard form and the given ordered triple. This time, let's choose the arbitrary values \( a = b = c = -2 \) and \( d = 12 + 1 + 7/2 \) for the first equation, and \( a = 3, b = 2, c = 1, d = -18 -1 -7/2 \) for the second one. This gives us a second set of equations:1. \( -2x -2y -2z = 12 +1 +7/2 = 11 \)2. \( 3x + 2y + z = -18 -1 -7/2 = -21 \)

Key Concepts

Ordered Triple

An ordered triple is a set of three numbers that represent the values of variables in a three-dimensional coordinate system. In the context of systems of linear equations, an ordered triple is usually written in the form \((x, y, z)\). Each value within the ordered triple corresponds to the solution for one of the variables in a system of equations. For instance, with the ordered triple \((-6, -\frac{1}{2}, -\frac{7}{4})\), \(x\) is \(-6\), \(y\) is \(-\frac{1}{2}\), and \(z\) is \(-\frac{7}{4}\). This plays a crucial role in verifying whether a set of linear equations forms a consistent system that possesses a unique solution.Working with ordered triples can help in illustrating solutions in spatial representations. When the ordered triple is a solution to a system of equations, it represents the point where the graphs of the equations intersect in three-dimensional space. Thus, it confirms if the equations provided truly have a common solution point.

Standard Form Equation

Standard form is a way of writing down equations, constructed in the form of \(ax + by + cz = d\). Here, \(a\), \(b\), and \(c\) are coefficients of the variables, while \(d\) is the constant term. The standard form equation is a fundamental representation in linear algebra for expressing straight lines or planes when three variables are involved.In forming systems of equations, the standard form is beneficial due to its structured approach, which simplifies the process of computation and verification. By maintaining a uniform format, it becomes easier to manipulate and solve equations using techniques such as substitution or elimination. Furthermore, using the standard form assists in easily aligning and comparing multiple equations to determine their relationships and intersection points.

Linear Algebra

Linear Algebra is a branch of mathematics that focuses on vectors, vector spaces, and linear equations. The subject is foundational in dealing with systems of equations and is essential for understanding and solving linear systems, especially those involving matrices and determinants. One of the core aspects of linear algebra is its application in solving systems of linear equations. These systems consist of equations where each represents a line (in two dimensions) or a plane (in three dimensions) and finding their intersection points. This is where ordered triples and standard form equations come into play, allowing us to express and solve these systems effectively. Techniques such as Gaussian elimination, matrix inverses, and Cramer's rule are tools derived from linear algebra principles that aid in finding solutions to these equations. Understanding linear algebra helps students visualize solutions, model complex systems, and develop skills that extend beyond mathematics into fields such as physics, engineering, and computer science. It embodies the procedures and techniques that allow us to handle and simplify complex relationships between multiple linear expressions.