Problem 23 - Elementary Algebra

Question

In the following problems, solve each of the conditional equations. $$ -3 k=126 $$

Step-by-Step Solution

Answer

Answer: k = -42

Step 1: Identify the equation

We are given the equation: $$ -3k = 126 $$

Step 2: Isolate the variable k

We need to isolate the variable k. To do this, we will divide both sides of the equation by the coefficient of k, which is -3: $$ k = \frac{126}{-3} $$

Step 3: Simplify the expression

Now, we can simplify the expression on the right side of the equation: $$ k = -42 $$

Step 4: Write the solution

The solution of the given conditional equation is: $$ k = -42 $$

Key Concepts

Conditional Equations

When working with linear equations, you may come across a specific type known as conditional equations. These equations are only true for certain values of the variable in question, unlike identities which hold true for all possible values. The given exercise \( -3k = 126 \) is a perfect example of a conditional equation, as it is only true when \( k \) takes on a specific value. To solve such an equation, one must find the particular value of the variable that satisfies the equation. This process is integral to understanding how different variables influence the outcome of mathematical problems and is widely applied across algebra.

Conditional equations are prevalent in solving real-world problems where conditions must be met, such as when balancing budgets, mixing solutions with precise concentrations, or even calculating rates of speed. Grasping these types of equations is essential, as they lay the groundwork for more advanced mathematical concepts and problem-solving strategies.

Variable Isolation

The heart of solving any linear equation lies in the skill of variable isolation. This technique involves manipulating the equation so that the variable we want to solve for stands alone on one side of the equal sign. In the given equation \( -3k = 126 \) from the exercise, variable isolation is achieved by dividing both sides of the equation by -3, the coefficient of \( k \). This step is crucial as it transforms the equation into a much more straightforward form: \( k = -42 \).

Isolating the variable is typically the second step after identifying the equation. It requires an understanding of inverse operations, such as adding the opposite or multiplying by the reciprocal, and is applicable not only in algebra but also in calculus, economics, physics, and other STEM fields. By mastering variable isolation, students can tackle more complex equations with confidence.

Equation Simplification

Completing the process of solving a linear equation, equation simplification is the step that follows variable isolation. In our exercise, once the variable \( k \) has been isolated, we arrive at the expression \( k = \frac{126}{-3} \). Equation simplification involves carrying out the necessary arithmetic operations to find the simplest form of the answer. In this case, we divide 126 by -3 to get \( k = -42 \).

Simplifying an equation makes interpretation and verification of solutions more manageable. It also ensures that answers are presented in their most reduced form, which is a standard mathematical practice. Simplification can involve combining like terms, reducing fractions, or applying properties of operations. Understanding how to simplify equations effectively allows students to tidy up their solutions, making them more presentable and easier to check for correctness.