Problem 17 - Elementary Algebra

Question

In the following problems, solve each of the conditional equations. $$ 6 p=-108 $$

Step-by-Step Solution

Answer

Answer: The value of p in the given equation is -18.

Step 1: Write down the given equation

Write down the equation we are trying to solve: $$ 6p = -108 $$

Step 2: Isolate the variable p

To isolate the variable p, we need to divide both sides of the equation by 6: $$ \frac{6p}{6} = \frac{-108}{6} $$

Step 3: Simplify and find the value of p

After dividing both sides by 6, we can simplify the equation to find the value of p: $$ p = -18 $$ So, the solution to the given conditional equation is \(p = -18\).

Key Concepts

Isolate the Variable

When working with algebraic equations, the first critical step is often to isolate the variable. This means you rearrange the equation to get the unknown variable by itself on one side of the equality sign. To do this, you perform operations that 'undo' whatever is being done to the variable.

In the exercise, the variable we want to isolate is represented by 'p', and the equation is 6p = -108. To isolate 'p', we need to divide both sides of the equation by 6, as each 'p' is being multiplied by 6. With division being the opposite operation of multiplication, we neutralize the multiplication by 6 and effectively get 'p' on its own.

Algebraic Equations

An algebraic equation is a statement that shows the relationship between different expressions that contain variables. These equations will typically have an 'equals' sign and at least one unknown value that needs to be found.

Our example equation, 6p = -108, has one unknown, 'p', and constants (6 and -108). The equation shows how 'p' is related to the constants. Remember, the goal in solving these equations is to find the value of the variable that makes the equation true.

Structure of Algebraic Equations

Understanding the structure of algebraic equations is paramount. These equations can vary greatly in complexity, from simple ones like ours to much more complicated multi-variable equations.

Simplifying Equations

After isolating the variable, it's often necessary to simplify the equation. This means reducing it to its simplest form to make it easier to understand or solve. Simplifying can include combining like terms, reducing fractions, or performing arithmetic operations.

Once we've isolated 'p' by dividing by 6, we then have p = -18. This is a simplified form of the equation, where 'p' is clearly defined without any extraneous information or operations. It's direct and to the point, showing the solution to the problem without anything left over.

Elementary Algebra

The exercises we've been looking at fall under the umbrella of elementary algebra. This area of mathematics provides the basis for more advanced studies in algebra. It involves the manipulations of algebraic expressions and solving of algebraic equations.

In elementary algebra, you learn the foundational skills needed to handle variables, constants, and the various algebraic operations (addition, subtraction, multiplication, and division). Mastery of these skills is fundamental to progress in mathematics as they are used to solve a wide range of practical and theoretical problems.