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Chapter 7: Problem 76

Solve each equation and check the result. If an equation has no solution, so indicate. $$ \frac{3}{p+6}-2=\frac{7}{p+6} $$

Short Answer

Expert verified
The solution is \( p = -8 \).

Step by step solution

01

Identify the Equation

The given equation is \( \frac{3}{p+6} - 2 = \frac{7}{p+6} \). We need to solve for \( p \).
02

Simplify the Equation

Notice that the fractions have the same denominator, \( p+6 \). Subtract the left fraction from the right fraction:\[\frac{3 - 7}{p+6} = 2\]This simplifies to:\[\frac{-4}{p+6} = 2\]
03

Solve for the Variable

Clear the fraction by multiplying both sides of the equation by \( p + 6 \):\[-4 = 2(p + 6)\]
04

Expand and Rearrange

Distribute the 2 on the right side:\[-4 = 2p + 12\]Subtract 12 from both sides:\[-4 - 12 = 2p\]This results in:\[-16 = 2p\]
05

Isolate the Variable

Divide both sides by 2 to solve for \( p \):\[p = -8\]
06

Check the Solution

Substitute \( p = -8 \) into the original equation to verify:\[\frac{3}{-8+6} - 2 = \frac{7}{-8+6}\]This simplifies to:\[\frac{3}{-2} - 2 = \frac{7}{-2}\]Calculating gives:\[-\frac{3}{2} - 2 = -\frac{7}{2}\]Convert \(-2\) to have a denominator of 2:\[-\frac{3}{2} - \frac{4}{2} = -\frac{7}{2}\]This identity confirms the solution is correct as both sides are equal.

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Key Concepts

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

Fraction Simplification
To tackle a rational equation like \( \frac{3}{p+6} - 2 = \frac{7}{p+6} \), it is important to understand fraction simplification. In this equation, both fractions have the same denominator, \( p+6 \). This similarity allows us to merge the fractions both on the right and the left by combining their numerators over a common denominator, leading to:
  • Left Numerator: 3 - Right Numerator: 7
  • Combined, it gives: \( \frac{3-7}{p+6} \)
  • Simplifying this, we get: \( \frac{-4}{p+6} \)
This simplification process is a powerful tool. It reduces complexity and sets us up to isolate the variable by transforming a rational equation into a linear equation.
Distributive Property
The distributive property is a useful algebraic tool. It allows you to simplify equations by multiplying through parentheses. Once we've combined like terms and simplified the fraction as \( \frac{-4}{p+6} = 2 \), we want to remove the denominator by multiplying both sides by \( p+6 \). This clears the fraction:
  • Equation: \(-4 = 2(p+6)\)
  • Apply Distributive Property: \(-4 = 2p + 12\)
This transforms the equation into a more familiar linear form. The use of distributive property is essential in transitioning from complex fractions to straightforward equations you can more easily solve.
Checking Solutions
Checking solutions ensures that your calculated answer is indeed correct. After determining the variable solution as \( p = -8 \), substituting back into the original equation verifies the result:
  • Substitute \( p = -8 \): \( \frac{3}{-8+6} - 2 = \frac{7}{-8+6} \)
  • Which simplifies to: \( \frac{3}{-2} - 2 = \frac{7}{-2} \)
  • Converting \(-2\) to have a denominator of 2: \( -\frac{3}{2} - \frac{4}{2} = -\frac{7}{2} \)
Both sides match, confirming that our solution \( p = -8 \) is correct. Always remember this verification step to avoid errors in your work.
Isolating Variables
Isolating variables is the crucial step when solving equations. After using the distributive property and simplifying, you arrive at an equation like \(-16 = 2p\). The goal is to get \( p \) by itself:
  • You divide both sides by 2, yielding \( p = -8 \).
This process of isolating the variable is fundamental. It transforms your equation to a form where the variable is independent, giving a clear solution. Practicing this skill will aid in solving not only rational equations but many other complex algebraic problems.

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Most popular questions from this chapter

Perform the operations. Simplify, if possible. $$ \frac{5}{x^{2}-9 x+8}-\frac{3}{x^{2}-6 x-16} $$

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