Problem 35 - Prealgebra

Question

Solve each equation. Check your solution. $$p-(-13.35)=-19.72$$

Step-by-Step Solution

Answer

The solution is \(p = -33.07\).

Step 1: Understand the Problem

You are given the equation \(p - (-13.35) = -19.72\). The goal is to solve for \(p\), which means finding the value of \(p\) that makes this equation true.

Step 2: Simplify the Equation

Notice that subtracting a negative is the same as adding the opposite. Therefore, the equation \(p - (-13.35) = -19.72\) can be rewritten as \(p + 13.35 = -19.72\).

Step 3: Isolate \(p\)

Subtract 13.35 from both sides of the equation to isolate \(p\). This results in \(p = -19.72 - 13.35\).

Step 4: Calculate the Solution

Perform the arithmetic on the right side of the equation: \(-19.72 - 13.35 = -33.07\). So, \(p = -33.07\).

Step 5: Check the Solution

Substitute \(p = -33.07\) back into the original equation to check the solution. Replace \(p\) in the equation: \(-33.07 - (-13.35) = -19.72\). Simplify the left side: \(-33.07 + 13.35 = -19.72\). Since both sides of the equation are equal, the solution is verified.

Key Concepts

Solving Linear Equations

Solving linear equations is a foundational skill in algebra. A linear equation is an equation that forms a straight line when graphed and typically looks like this: \( ax + b = c \). The goal is to find the unknown variable, often represented as \( x \) or another letter. To solve for the unknown:

• Isolate the variable: To find out what the variable represents, you need to isolate it on one side of the equation. This is often done by using inverse operations such as addition, subtraction, multiplication, or division.
• Balance the equation: Whatever you do to one side of the equation, do the same to the other to keep it balanced. This is crucial for maintaining equality.

Let's consider the example given: \( p - (-13.35) = -19.72 \). Here, you need to isolate \( p \). By recognizing that subtracting a negative is the same as adding a positive, it simplifies the task right away. Always remember to check your solution by substituting it back into the original equation.

Understanding Negative Numbers

Negative numbers might seem a little confusing at first. They are numbers with a value less than zero, often used to represent things like debt or temperatures below freezing. When dealing with negative numbers in equations, keep these points in mind:

• Subtracting a negative: As in the example \( p - (-13.35) \), subtracting a negative is equivalent to adding its positive counterpart, so the expression becomes \( p + 13.35 \).
• Negative results: If both numbers involved in the operation are negative, the result can also be negative, as illustrated by the arithmetic \( -19.72 - 13.35 = -33.07 \).

Working through problems with negatives becomes easier with practice. Always pay attention to signs, and double-check calculations to avoid common errors.

Performing Arithmetic Operations

Arithmetic operations are the basic math processes including addition, subtraction, multiplication, and division. In the context of solving equations, they help manipulate and simplify expressions. In our example, let's see how they come into play:

• Addition and subtraction: These are the most common operations when solving equations. Adding or subtracting helps in isolating the variable, as shown by adding 13.35 to both sides of the equation to simplify \( p - (-13.35) = -19.72 \).
• Order of operations: Remember the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction) as a guide to know which operation to perform first in solving more complex equations.

For the calculation \( -19.72 - 13.35 = -33.07 \), make sure each step follows the correct usage of arithmetic rules, especially when dealing with subtraction and negative numbers. This ensures the final solution is accurate and verified.