Problem 1 - A Graphical Approach to College Algebra

Question

Verify that the given ordered triple is a solution of the system. Do not use a calculator. \((-3,6,1)\) \begin{aligned}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 to the system of equations.

Step 1: Insert the ordered triple into the first equation

Given the first equation \(2x + y - z = -1\), insert the values from the ordered triple \((-3, 6, 1)\) into this equation. This means substituting \(x = -3\), \(y = 6\), and \(z = 1\). The equation becomes: \[2(-3) + 6 - 1 = -1.\] Simplify the equation: \[-6 + 6 - 1 = -1.\] Therefore, \(-1 = -1\), which verifies the ordered triple is a solution to the first equation.

Step 2: Insert the ordered triple into the second equation

Using the second equation \(x - y + 3z = -6\), substitute \(x = -3\), \(y = 6\), and \(z = 1\). The equation becomes: \[-3 - 6 + 3(1) = -6.\] Simplify this to: \[-3 - 6 + 3 = -6.\] Therefore, \(-6 = -6\), which verifies the ordered triple is a solution to the second equation.

Step 3: Insert the ordered triple into the third equation

For the third equation \(-4x + y + z = 19\), substitute \(x = -3\), \(y = 6\), and \(z = 1\). Substitute to get: \[-4(-3) + 6 + 1 = 19.\] Simplify the equation: \[12 + 6 + 1 = 19.\] Therefore, \(19 = 19\), which verifies that the ordered triple is a solution to the third equation.

Key Concepts

Understanding Ordered Triples

An ordered triple is a set of three numbers used to represent a point in three-dimensional space. In our case, the ordered triple is \((-3, 6, 1)\). Each number in this set corresponds to a variable in the system of equations:

• The first number is \(-3\), which we'll assign to \(x\).
• The second number is \(6\), which will represent \(y\).
• The third number is \(1\), which is for \(z\).

Ordered triples are useful for identifying specific points in a 3D coordinate space. In the context of a system of equations, they offer a potential solution where the variables meet certain conditions simultaneously. For verification, we substitute these values into each equation in the system to see if they satisfy all of them.

The Role of Algebraic Verification in Systems of Equations

Algebraic verification is the process of checking whether a proposed solution satisfies all the equations within a system. For this, each equation is assessed individually. Let's consider the first equation of the given system, \(2x + y - z = -1\).
To verify the solution \((-3, 6, 1)\), substitute these values into the equation. You'll replace \(x\) with \(-3\), \(y\) with \(6\), and \(z\) with \(1\).
This leads us to:

• \[2(-3) + 6 - 1 = -1\]
• Which simplifies to \[-6 + 6 - 1 = -1\]
• And this confirms \,\(-1 = -1\)\, verifying the solution with this equation.

Engaging in algebraic verification ensures each component of the solution satisfies the respective conditions imposed by all equations involved in the system.

Solution Verification Process

Solution verification refers to the sequential substitution of the proposed values into each equation to verify their correctness. In any system of equations, as we have here, multiple equations describe relationships between different variables.
To confirm \((-3, 6, 1)\) as a solution, we insert these values into each of the system's equations.
Once substituted, each equation should return true, meaning both sides of the equation should balance. Consider the third equation in our example, \(-4x + y + z = 19\). When we substitute our ordered triple, the expression becomes:

• \[-4(-3) + 6 + 1 = 19\]
• Which simplifies to \[12 + 6 + 1 = 19\]
• And checks out with\, \,\( 19 = 19\)\.

This methodical plug-and-check approach ensures each solution's validity, intercepting errors or mathematical oversight.

Solving Using the Substitution Method

The substitution method is one way to solve a system of equations. Here’s how it generally works, although in our exercise the solutions were simply verified rather than solved this way:
Start by solving one of the equations for one variable in terms of the others. This provides an expression you can substitute into the other equations.
For example, suppose we shape an equation around one variable's expression and then use this to replace that variable in the other equations.

• Take the equation \(x = y - 3z\) (hypothetical for demonstration)
• Substitute this expression anywhere \(x\) appears in the other equations.
• Repeat this technique proportionately until all settings portray expressions involving fewer variables until you hit single-variable equations.

Such operations transform multi-variable conundrums into single-variable challenges, making the analyzing and solving process moderately linear and more straightforward.