Problem 50 - College Algebra

Question

Make Sense? In Exercises \(48-51\), determine whether each statement makes sense or does not make sense, and explain your reasoning. I'm solving a three-variable system in which one of the given equations has a missing term, so it will not be necessary to use any of the original equations twice when I reduce the system to two equations in two variables.

Step-by-Step Solution

Answer

The given statement does not make sense systematically, as even with one equation having a missing term, it might still be necessary to use the original equations more than once for reducing the system.

Step 1: Interpreting the Statement

It is essential to break down the assumption stated in the word problem. According to the text, the claim is that when reducing a three-variable system to two variables, even if one equation has a missing term, it will not be obligatory to use any of the original equations twice.

Step 2: Analyzing the Statement

Upon reduction, a three-variable system to two variables usually requires using one equation multiple times. However, if an equation has a term missing (essentially making it already a two-variable equation), it could potentially streamline the solution process, but it does not guarantee that original equations are not needed to be used twice for complete reduction.

Step 3: Concluding the Statement

So, it can be concluded that the statement provided is not necessarily true. Even when one of the equations in a three-variable system lacks a term (and is thus a two-variable equation), it may still be required to use one of the other original equations again in order to fully reduce the system into two variables.

Key Concepts

System of Equations

A system of equations is a collection of two or more equations with a set of variables. The goal is to find values for the variables that satisfy all the equations concurrently. In algebra and mathematics fields, systems can have two variables, known as a two-variable system, or even more, like a three-variable system.

When dealing with a three-variable system, you have equations like:

• \( ax + by + cz = d \)
• \( ex + fy + gz = h \)
• \( ix + jy + kz = l \)

Here, \(x, y, \text{and} \, z\) are the variables, and the goal is to determine their values. A unique aspect of these systems is that they may return a unique solution, infinite solutions, or no solution at all.

Understanding how these equations relate helps in solving them, particularly with techniques like substitution or elimination. By grasping the connection among equations, you can efficiently reduce complexity and find the correct solutions.

Variables in Algebra

In algebra, variables are symbols used to represent unknown values. Common variables used include \(x, y, \text{and} \, z\), especially in three-variable systems. Think of them as placeholders, waiting to be defined by solving the equations in which they exist.

The characteristics of variables:

• They allow generalization; you can solve for any value rather than just a fixed number.
• They help model real-world problems by converting them into solvable equations.
• They maintain consistency within a problem, representing the same unknown throughout the equations.

In the context of a three-variable system, these variables interact within the equations. Successfully figuring out their values means you have solved the system. Thus, understanding the role variables play is crucial.

Reducing Equations

Reducing equations within a system involves simplifying the problem to make it more manageable. This is often done by eliminating one or more variables, transforming a complex three-variable system into a simpler one or two-variable system.

Key strategies for reducing equations:

• Substitution: Solve one equation for a variable, then substitute that into other equations to simplify.
• Elimination: Combine equations to remove a variable, leading to a reduced system.

Reducing helps clarify the problem, allowing for a more straightforward path to solution. Even when one equation lacks a term, indicating it's already a two-variable equation, careful reduction remains vital. Reduction does not always ensure negotiation through fewer steps, but it aids greatly in understanding the structure of the system.

Solution Process

The solution process for a three-variable system is strategic and involves several steps:

• Identify and label: Clearly define all equations and known variables.
• Choose a method: Decide whether to use substitution, elimination, or graphing based on the equations' structure and complexity.
• Reduce and solve: Reduce the system to two variables, solve for one or both, then backtrack to find the third variable.
• Verify: Always check your solution by substituting back into the original equations to ensure correctness. A valid solution should satisfy all original equations.

In the given scenario, even if an equation misses a term, it doesn't simplify the process to the extent of avoiding the original ones. Verify whether the equations are reused or if a unique path can be applied each time. This structured approach assures a correct solution.