Problem 4 - College Algebra

Question

In Exercises \(1-4,\) determine if the given ordered triple is a solution of the system. $$ \begin{aligned} &(-1,3,2)\\\ &\left\\{\begin{array}{c} x-2 z=-5 \\ y-3 z=-3 \\ 2 x-z=-4 \end{array}\right. \end{aligned} $$

Step-by-Step Solution

Answer

Yes, the ordered triple (-1, 3, 2) is a solution to the system of equations.

Step 1: Substitute values in the first equation

The first equation is \(x - 2z = -5\). Let's substitute \(x = -1\) and \(z = 2\) into the equation. We get \(-1 - 4 = -5\), which simplifies to \(-5 = -5\). This is correct and hence, satisfies the first equation.

Step 2: Substitute values in the second equation

The second equation is \(y - 3z = -3\). Substituting \(y = 3\) and \(z = 2\) into the equation results in \(3 - 6 = -3\), or \(-3 = -3\). This is correct and satisfies the second equation.

Step 3: Substitute values in the third equation

The third equation is \(2x - z = -4\). Substituting \(x = -1\) and \(z = 2\) into the equation results in \(-2 - 2 = -4\), or \(-4 = -4\). This is also correct and satisfies the third equation.

Key Concepts

Understanding Ordered Triples

An "ordered triple" refers to a specific group of three numbers in a defined order, typically represented as \((x, y, z)\). These numbers correspond to the variables in a system of three linear equations. It is essential because each number in the triple must satisfy the respective variable's position in every equation of the system.
For example, in the given problem, the ordered triple \((-1, 3, 2)\) corresponds to \(x = -1\), \(y = 3\), and \(z = 2\). By understanding this structure, we can determine if this specific set of numbers solves all equations in the system. Simply put, the ordered triple provides a succinct way to check each variable's assignment across multiple equations.

Solution Verification Simplified

Verifying a solution means checking if an ordered triple satisfies all equations in the system. This is a crucial step to ensure that our proposed solution truly works.
Listed below is how solution verification proceeds:

• Substitute each value from the ordered triple into its respective variables in the equations.
• Perform arithmetic operations to simplify both sides of the equations.
• Check if both sides of each equation are equal post-substitution.

In our example, we substituted \((-1, 3, 2)\) into each equation to check if the left-hand side equals the right-hand side. This step-by-step substitution proves if the ordered triple fits or not, enhancing our understanding of the solution's accuracy.

The Substitution Method Explained

The substitution method involves replacing a variable with a value or expression to simplify and solve equations. It is a handy tool when dealing with systems of equations where finding a direct solution might be cumbersome.
Let's see how it operates in our context:

• Select a value for a variable from the ordered triple.
• Replace that variable in one of the equations using this value. Simplify the equation to see if it balances.
• Repeat for the remaining variables and equations.

In the given system, each variable from the ordered triple \((-1, 3, 2)\) was substituted into their respective equations. This demonstrated whether each equation held true. Substitution simplifies solving systems by turning complex variable interactions into manageable calculations.

Linear Equations in Systems

Linear equations are algebraic expressions where each term is either a constant or a product of a constant with a single variable. In these equations, variables are not raised to any power other than one. Systems of linear equations involve multiple linear equations working together.
Here's a brief look at their role:

• Each linear equation provides a geometric line in space if graphed.
• The solution to the system is the point(s) where these lines intersect in a 3D space, represented by an ordered triple.

In our exercise, three linear equations form a system, with the ordered triple \((-1, 3, 2)\) potentially representing their intersection point. Understanding how linear equations form systems allows us to predict and verify solutions, visualizing how different equations can converge to a single solution point.