Problem 96 - Algebra 1

Question

Solve the equation. \(-\frac{p}{7}=-9\)

Step-by-Step Solution

Answer

The solution for the equation \(-\frac{p}{7} = -9\) is \(p = 63\).

Step 1: Isolate the variable

In the equation \(-\frac{p}{7} = -9\), multiply both sides by \(-7\) to isolate the variable. This gives: \(p = -9 \times -7\).

Step 2: Simplify the equation

By multiplying \(-9\) by \(-7\), the equation simplifies to: \(p = 63\).

Key Concepts

Understanding Equations

An equation is a mathematical statement where two expressions are equal to each other, typically separated by an equal sign, like \( a = b \). In our exercise, we have \( -\frac{p}{7} = -9 \). This equation tells us that the expression on the left, \( -\frac{p}{7} \), evaluates to the same number as the expression on the right, which is \(-9\).
To solve an equation, our goal is to find the value of the variable, which makes both sides of the equation equal. The variable in this problem is \( p \). By finding \( p \), we will satisfy the condition stated by the equation. Understanding equations is important because they allow us to express relationships between numbers and variables, and they are fundamental in solving countless real-world problems.

Variable Isolation Basics

Variable isolation is a key step in solving equations, where we aim to rearrange the equation so that the variable we are interested in is by itself on one side of the equation. For this problem, we start with \( -\frac{p}{7} = -9 \). Here the variable is \( p \), and it is part of a division operation. To isolate \( p \), we need to get rid of the \(-\frac{1}{7}\) coefficient.

• First, understand the goal: get \( p \) alone.
• Consider the operation pairing: since \( p \) was divided by \(-7\), you need to multiply by \(-7\) to reverse it. This is because division and multiplication are inverse operations.

Multiplying both sides by \(-7\) removes the fraction: \[ -7 \times \left( -\frac{p}{7} \right) = -7 \times (-9)\]The left side simplifies to just \( p \), leaving the equation \( p = 63 \). Now, \( p \) is isolated, and we've found the value that makes both sides equal.

Working with Negative Numbers

Negative numbers can sometimes be confusing, but they follow their own set of rules that are consistent. In this exercise, both numerical values and the variable expression involve negative numbers. We start by looking at \(-\frac{p}{7} = -9\), where both the divisor and the number on the right side are negative.

• When multiplying two negative numbers, the result is positive. Hence, \(-9 \times -7 = 63\).
• If you had added a negative number to another, you'd just subtract them.

Using these rules ensures precision when handling negative numbers. Therefore, always apply these principles while solving equations and simplifying expressions involving negatives to arrive at the correct solution.Embrace the idea that negatives are not tricksters—they're a part of arithmetic's uniform rules that help balance equations like those you see here.