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Chapter 2: Problem 49

Solve equation, and check your solution. \(\frac{2}{7} p=4\)

Short Answer

Expert verified
The solution is \(p = 14\).

Step by step solution

01

Understand the Equation

The given equation is \(\frac{2}{7} p = 4\). It states that \(\frac{2p}{7}\) equals 4.
02

Isolate the Variable

To solve for \(p\), multiply both sides of the equation by 7. This will eliminate the denominator: \(\frac{2}{7} p \times 7 = 4 \times 7\).
03

Simplify

Simplify both sides to isolate \(p\): \(2p = 28\).
04

Solve for p

Divide both sides by 2 to solve for \(p\): \(2p/2 = 28/2\), which simplifies to \(p = 14\).
05

Check the Solution

Substitute \(p = 14\) back into the original equation to verify: \(\frac{2}{7} \times 14 = 4\). This simplifies to \(4 = 4\), confirming the solution is correct.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Isolating Variables
When solving linear equations, one crucial step is isolating the variable you are solving for. This involves rearranging the equation so that the variable appears alone on one side of the equation. In the example \(\frac{2}{7} p = 4\), our goal is to get \(p\) by itself. To do this, we first eliminate the fraction by multiplying both sides by the denominator. This step is essential because it simplifies the equation into a form that is easier to work with. Remember: whatever operation you do to one side of the equation, you must do to the other side to maintain equality.
Multiplication Property of Equality
The Multiplication Property of Equality states that when you multiply both sides of an equation by the same nonzero number, the two sides remain equal. In the example, we use this property to remove the fraction by multiplying both sides by 7: \[ \frac{2}{7} p \times 7 = 4 \times 7 \]. This action cancels the \( \frac{1}{7} \) on the left side, leaving us with \(2p\) on one side of the equation and simplifying the right side to 28. This property is a fundamental tool in solving linear equations as it allows us to manipulate the equation without changing its solution.
Verifying Solutions
After isolating the variable and solving the equation, it's essential to verify the solution. This step ensures that our solution satisfies the original equation. To verify, substitute the value of the variable back into the original equation and check if both sides are equal. For instance, if we find that \( p = 14 \) from solving \( \frac{2}{7} p = 4 \), we substitute \( 14 \) back into the original equation: \[ \frac{2}{7} \times 14 = 4 \]. Simplifying both sides, we find that \( 4 = 4 \), confirming our solution is correct. Verification is a crucial step to validate our answer.

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