Problem 4 - A Graphical Approach to Precalculus with Limits

Question

Verify that the given ordered triple is a solution of the system. Do not use a calculator. $$\begin{aligned} (-1,-2,-3) & \\ x-y+z=&-2 \\ x-2 y+z=& 0 \\ y-z=& 1 \end{aligned}$$

Step-by-Step Solution

Answer

The ordered triple \((-1, -2, -3)\) is a solution to the system.

Step 1: Introduction to the problem

We need to verify if the ordered triple \((-1, -2, -3)\) is a solution to the given system of equations. This means substituting \(x = -1\), \(y = -2\), and \(z = -3\) into each equation and checking if they hold true.

Step 2: Verify first equation

Substitute \(x = -1\), \(y = -2\), and \(z = -3\) into the first equation: \(x - y + z = -2\).\\((-1) - (-2) + (-3) = -2\).\Simplify: \(-1 + 2 - 3 = -2\).\The left-hand side simplifies to \(-2\), which equals the right-hand side.

Step 3: Verify second equation

Substitute \(x = -1\), \(y = -2\), and \(z = -3\) into the second equation: \(x - 2y + z = 0\).\\((-1) - 2(-2) + (-3) = 0\).\Simplify: \(-1 + 4 - 3 = 0\).\The left-hand side simplifies to \(0\), which equals the right-hand side.

Step 4: Verify third equation

Substitute \(y = -2\) and \(z = -3\) into the third equation: \(y - z = 1\).\\(-2 - (-3) = 1\).\Simplify: \(-2 + 3 = 1\).\The left-hand side simplifies to \(1\), which equals the right-hand side.

Step 5: Conclusion

Since substituting \((-1, -2, -3)\) into all three equations results in true statements, this ordered triple is indeed a solution to the system of equations.

Key Concepts

Ordered Triple

In the context of systems of linear equations, an ordered triple represents a set of three numbers, typically written in parentheses like \((-1, -2, -3)\). Each number in this triple corresponds to a specific variable in a system of equations, such as \((x, y, z)\). Here's how the ordered triple works:

• It's a solution if the numbers satisfy all the equations in the system when substituted for their respective variables.
• In our example, \((-1, -2, -3)\) means \x = -1\, \y = -2\, and \z = -3\.
• Satisfying all the equations validates that the ordered triple is indeed a solution.

Ordered triples are especially useful in three-variable systems like the one we are considering. Important to note is that all three values must work simultaneously in each equation for it to be a valid solution.

Solution Verification

Solution verification is a crucial process in mathematics, particularly when dealing with systems of equations. Here's how it works:

• The goal is to confirm if a proposed solution, such as an ordered triple, satisfies all the given equations.
• This is done by substituting each component of the ordered triple into every equation of the system.
• The result of each substitution must verify the original equation, leading to true statements on both sides of the equation.

For example, if you plug \(-1\), \(-2\), and \(-3\) respectively into the equations \x - y + z = -2\, \x - 2y + z = 0\, and \y - z = 1\, each should equate meaning both sides are equal after simplification. This confirmed that \(-1, -2, -3\) is a bona fide solution of the system, verifying that the system holds true for the proposed ordered triple.

Linear Equations

Linear equations are the backbone of solving problems involving systems of equations. A linear equation is an equation of the first degree, meaning its variables are to the power of one. In our discussed problem, the system consists of linear equations:

• Each equation can be represented in a linear form, for instance, \x - y + z = -2\.
• These are equations that form straight lines when graphed, each representing different planes in a three-dimensional space.
• The intersection point(s) of these planes, if they exist, represent the solution of the system.

Linear equations are easy to handle because they do not contain exponents or nonlinear terms. They can include more than one variable, as seen in the system of three equations touching the variables \(x\), \(y\), and \(z\). Understanding them is pivotal to advancing in solving more complex mathematical systems.