Problem 1 - 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. $$\begin{aligned} (-3,6,1) & \\ 2 x+y-z=&-1 \\ x-y+3 z=&-6 \\ -4 x+y+z=& 19 \end{aligned}$$

Step-by-Step Solution

Answer

The ordered triple \((-3, 6, 1)\) is a solution of the system.

Step 1: Understand the Problem

We need to verify if the ordered triple \((-3, 6, 1)\) is a solution of the given system of equations. This means substituting \(x = -3\), \(y = 6\), and \(z = 1\) into each equation and checking if all the equations hold true.

Step 2: Substitute into the First Equation

Substitute \(x = -3\), \(y = 6\), and \(z = 1\) into the first equation: \[2x + y - z = -1\]That gives us:\[2(-3) + 6 - 1 = -1\]Calculate:\[-6 + 6 - 1 = -1\]So the expression simplifies to:\[-1 = -1\] This equation holds true.

Step 3: Substitute into the Second Equation

Substitute \(x = -3\), \(y = 6\), and \(z = 1\) into the second equation:\[x - y + 3z = -6\]That gives us:\[-3 - 6 + 3(1) = -6\]Calculate:\[-3 - 6 + 3 = -6\]Simplify:\[-6 = -6\] This equation also holds true.

Step 4: Substitute into the Third Equation

Substitute \(x = -3\), \(y = 6\), and \(z = 1\) into the third equation:\[-4x + y + z = 19\]That gives us:\[-4(-3) + 6 + 1 = 19\]Calculate:\[12 + 6 + 1 = 19\]Simplify:\[19 = 19\] This equation holds true as well.

Step 5: Conclusion

Since the ordered triple \((-3, 6, 1)\) satisfies all three equations in the system, it is indeed a solution of the system.

Key Concepts

Linear Equations

Linear equations form the backbone of algebra. They are equations where each term is either a constant or a product of a constant and a single variable. These are characterized by having the highest degree of the variable as one. Linear equations can be represented in the form:

• Standard form: \(ax + by + cz = d\)
• Slope-intercept form: \(y = mx + c\), mainly for two variables

In this context, each equation represents a plane in three-dimensional space when formulated with three variables: \(x, y, \text{and } z\). These equations are key in solving systems of equations like the one in this exercise. By determining where these planes intersect, you can find the solution to the system. This solution is typically expressed as an ordered triple.

Ordered Triple

An ordered triple is a set of three numbers that define a point in three-dimensional space. Much like ordered pairs (used in two dimensions), an ordered triple consists of three elements written in a specific order: \((x, y, z)\). For this problem, the ordered triple \((-3, 6, 1)\) represents a potential solution where:

• \(x = -3\)
• \(y = 6\)
• \(z = 1\)

In the context of a system of equations, these values are substituted into each equation to verify if they satisfy all the equations. If the right-hand side equals the left-hand side for all equations when the values are substituted, the ordered triple is a solution to the system. This shows the point where the planes represented by the equations intersect.

Substitution Method

The substitution method is a technique used to solve systems of equations. It involves solving one of the equations for one variable and then substituting this expression into the other equations. Although substitution here is adapted for verification purposes, it typically follows these steps:

• Solve one equation for one of the variables (e.g., \(x\) in terms of \(y\) and \(z\))
• Use this expression wherever this variable appears in the other equations
• Solve the resulting equations for another variable

In the exercise, we substituted known values directly from the ordered triple into each equation. This confirms whether the ordered triple satisfies each equation, verifying that it is indeed a solution. It simplifies verifying if the equations are consistent with the values provided.

Solution Verification

Solution verification is the process of checking whether a proposed solution satisfies all the equations in a system. It's like getting the answer to a test and checking each step to make sure it's correct. For a system with three equations, you must confirm that the ordered triple satisfies all:

• Substitute the triple values into each equation
• Simplify to verify that both sides of the equation are equal
• Document that all equations uphold the integrity of the triple

In our example, every equation, upon substitution, balances correctly (e.g., \(-1 = -1\)), indicating that the given ordered triple \((-3, 6, 1)\) is a valid solution. This ensures accuracy and consistency, providing confidence that the system's solution has been correctly applied and verified.