Problem 5 - A Graphical Approach to College Algebra

Question

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

Step-by-Step Solution

Answer

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

Step 1: Verify the First Equation

Substitute the values of the ordered triple \((-2, -1, 3)\) into the first equation: \[x - y + z = 2\]Substitute \(x = -2\), \(y = -1\), and \(z = 3\):\[-2 - (-1) + 3 = 2\]Calculate the left side:\[-2 + 1 + 3 = 2\]\[2 = 2\]The first equation is satisfied.

Step 2: Verify the Second Equation

Substitute the values of the ordered triple \((-2, -1, 3)\) into the second equation:\[3x - 2y + z = -1\]Substitute \(x = -2\), \(y = -1\), and \(z = 3\):\[3(-2) - 2(-1) + 3 = -1\]Calculate the left side:\[-6 + 2 + 3 = -1\]\[-1 = -1\]The second equation is satisfied.

Step 3: Verify the Third Equation

Substitute the values of the ordered triple \((-2, -1, 3)\) into the third equation:\[x + y = -3\]Substitute \(x = -2\) and \(y = -1\):\[-2 + (-1) = -3\]Calculate the left side:\[-2 - 1 = -3\]\[-3 = -3\]The third equation is satisfied.

Key Concepts

Understanding Ordered Triples

An ordered triple is a set of three numbers in a specific sequence. It is typically expressed as \((x, y, z)\) and represents a point in three-dimensional space. Each coordinate corresponds to one dimension:

• \(x\): The position along the x-axis (usually horizontal).
• \(y\): The position along the y-axis (usually vertical).
• \(z\): The position along the z-axis (usually depth).

In the context of a system of equations, an ordered triple like \((-2, -1, 3)\) represents potential solutions for each of the variables in the equations. When verifying a solution, each coordinate from the ordered triple is substituted into the corresponding variables of the equations in the system. If all the equations turn out to be true, the ordered triple is a valid solution.
This provides a way to think of solutions in not just a two-dimensional plane (like pairs in 2D), but in a 3D space, granting more complexity and representing scenarios that might involve three different parameters at once.

Solution Verification Process

Verification of a solution to a system of equations involves checking each individual equation with the proposed solution. Here, we were given the ordered triple \((-2, -1, 3)\) to verify:

• **Substitution**: Each part of the ordered triple is plugged into the corresponding variable of each equation.
• **Calculation**: After substitution, you evaluate the equation to check if it holds true. This often involves performing simple arithmetic to see if both sides of the equation are equal.
• **Conclusion**: If all equations are satisfied, the ordered triple is confirmed as a solution.

Steps involve substituting \((-2, -1, 3)\) into the given equations: each verification step confirms a true statement like \(2 = 2\), \(-1 = -1\), and \(-3 = -3\). This method ensures a hands-on understanding of how given values are solutions through straightforward algebraic proving, building confidence in both logic and computation.

The Role of Algebraic Substitution

Algebraic substitution is a key technique in verifying solutions for systems of equations. It transforms the abstract numbers into use specific contexts. When applied, the procedure is straightforward:

• Identify the variables in each equation.
• Substitute the given values from the ordered triple into their respective places.
• Solve the resulting arithmetic to see if it holds true (achieves a true statement like \(a = a\)).

Algebraic substitution not only confirms the solution but enhances the understanding of how each part of the solution affects the whole. In doing so, it uses known values to validate the relationships expressed by the equations. This way, substitution assists in conceptualizing multiple variables interacting in a system of equations and acting as a practical tool to simplify complex problems into manageable calculations.