Question

Find two systems of equations that have the ordered triple as a solution. (There are many correct answers.) $$(1,-5,-3)$$

Short answer

Two possible systems of equations that have the ordered triple (1,-5,-3) as a solution are: x + y + z = -7 and 2x - y - 3z = 9.

Step-by-step solution

Creating the First Equation

In order to create an equation that has the solution \((1,-5,-3)\), we can start with a general form of a linear equation which is: \(ax + by + cz = d\), where \(x\), \(y\), and \(z\) are variables, and \(a\), \(b\), \(c\), and \(d\) are constants. As we know the solutions, let's try a simplest case by setting \(a=1\), \(b=1\), \(c=1\), and \(d=-7\). Our equation therefore is \(x + y + z = -7\). If we plug in the solution, we get \(1 + (-5) + (-3) = -7\), which verifies our equation.

Creating the Second Equation

We'll again start with the general form of a linear equation: \(ax + by + cz = d\). This time, let's change the values of the constants. Let's set \(a=2\), \(b=-1\), \(c=-3\), and \(d=9\). This gives us the second equation as \(2x - y - 3z = 9\). If we substitute the solution into this equation, we get \(2*1 - (-5) - 3*(-3) = 9\), which verifies this equation.

Key concepts

Ordered Triple

An ordered triple, in the context of systems of equations, is a set of three numbers that represent the solution to a system of three linear equations. It's written in the form \( (x, y, z) \) where \(x\), \(y\), and \(z\) correspond to the variables in the equations. The order of these numbers is crucial—as the name suggests—because each number corresponds to a specific variable and position in the equations. When you plug these numbers into each equation of the system, they should satisfy all the equations simultaneously.

For example, consider the ordered triple \( (1, -5, -3) \) from our exercise. Here, \(1\) is the value of \(x\), \( -5\) is the value of \(y\), and \( -3\) is the value of \(z\). A correct system of equations must result in true statements when these values are substituted for their respective variables. This specific ordered triple tells us that, whatever our equations may be, when we replace \(x\) with \(1\), \(y\) with \( -5\), and \(z\) with \( -3\), the equations should hold true. This brings us to our next concept, the linear equation.

Linear Equation

A linear equation is a type of equation that represents a straight line when graphed on a coordinate plane. In three variables, it takes the form \( ax + by + cz = d \) where \(a\), \(b\), and \(c\) are the coefficients of the variables \(x\), \(y\), and \(z\), respectively, and \(d\) is the constant term. The equation is called 'linear' because the highest power of any variable is one.

In the context of the given exercise, we created linear equations that would be satisfied by the ordered triple \( (1, -5, -3) \). By selecting different coefficients and constants \(a\), \(b\), \(c\), and \(d\), we can create a multitude of linear equations that satisfy this ordered triple. The underlying principle is that the left side of the equation must equate to the right side when the values from the ordered triple are substituted.

Understanding how to construct these equations is crucial in algebra, as it forms the foundation for modelling real-world situations with mathematical expressions. As we move forward, we'll explore how these individual equations come together to form algebraic solutions when encapsulated within a system.

Algebraic Solutions

Algebraic solutions in the realm of systems of equations refer to the set of values that satisfy all equations in the system simultaneously. When dealing with three variables, an algebraic solution is represented as an ordered triple, like the one we have discussed. To find these solutions, we can use various methods such as substitution, elimination, or matrix operations.

Our exercise focuses on creating systems of equations with a known solution, which means we design the equations with our ordered triple in mind. However, when tasked with finding solutions without known results, the process usually involves more complex steps and a deep understanding of manipulating equations. You might 'juggle' the variables around, eliminate them, substitute one variable for another, or even use graphical methods to see where the equations intersect, representing the solution where all the equations in the system agree.

In summary, algebraic solutions are the 'answers' we seek in a system of equations. They provide the precise values for variables that make the equations true, and understanding how to arrive at these solutions is a fundamental skill in algebra that extends to higher mathematics and real-world problem solving.