Problem 18 - Elementary Algebra

Question

In the following problems, solve each of the conditional equations. $$ 12 q=-180 $$

Step-by-Step Solution

Answer

Answer: The value of q in the given equation is -15.

Step 1: Identify the equation

We are given the equation: $$ 12q = -180 $$

Step 2: Divide both sides by 12

To isolate q, we need to divide both sides of the equation by 12: $$ \frac{12q}{12} = \frac{-180}{12} $$

Step 3: Simplify the equation

After dividing both sides by 12, we will get the value of q: $$ q = -15 $$ The solution to the given conditional equation is q = -15.

Key Concepts

Conditional Equations

Conditional equations are a specific type of equation in algebra that have particular solutions. Unlike identities, which are true for all values of the variables involved, conditional equations are true only under certain conditions. In this exercise, the given equation \(12q = -180\) is an example of a conditional equation. It is true only when the variable \(q\) is equal to -15.

Often, students may confuse conditional equations with other types of algebraic statements. It’s important to remember:

• Identity: True for all values, like \(x + 0 = x\).
• Conditional: True for specific values, like \(12q = -180\) is only true when \(q = -15\).

Understanding this distinction helps in determining the solution approach when faced with different algebraic problems.

Algebraic Manipulation

Algebraic manipulation is a toolkit of methods that we use to solve equations and simplify expressions. It involves using operations like addition, subtraction, multiplication, and division to isolate and find the value of a variable. In the context of our exercise, we use algebraic manipulation to solve for the variable \(q\) in the equation \(12q = -180\).

The key steps involved are:

• Isolate the variable: Perform operations that get the variable on one side of the equation alone.
• Perform opposite operations: Use the inverse operation to undo whatever is attached to the variable. For example, here \(q\) was multiplied by 12, so we performed division.

Effective algebraic manipulation allows us to handle more complex equations as well, paving the way for more advanced algebraic topics.

Division in Equations

Division plays a crucial role when solving equations, particularly conditional ones. It is often used when a variable is multiplied by a coefficient, as seen in the equation \(12q = -180\). Here, dividing both sides of the equation by 12 was a strategic move to isolate \(q\).

Important tips to remember include:

• Maintain balance: What you do to one side of the equation, you must also do to the other side. This keeps the equation true.
• Zero is special: Never divide by zero, as this is undefined in mathematics.
• Simplification: After division, simplify as much as possible to get the variable alone, leading to the solution.

Mastering division and understanding its role in equations like \(12q = -180\) will help you tackle a wide variety of algebraic equations with confidence.