/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 31 Solve the equations. $$ \fra... [FREE SOLUTION] | 91影视

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Chapter 5: Problem 31

Solve the equations. $$ \frac{6 a}{-5}=11 $$

Short Answer

Expert verified
Question: Solve the equation $\frac{6a}{-5} = 11$ for the variable 'a'. Answer: $a = \frac{-55}{6}$

Step by step solution

01

Clear the fraction by multiplying both sides with the denominator

To remove the fraction, we'll multiply both sides of the equation by -5: $$ (-5)\times\frac{6a}{-5}= 11\times(-5) $$
02

Simplify and isolate 'a'

After multiplying, the fraction will simplify, and we can solve for 'a': $$ 6a = -55 $$ Now, divide both sides by 6 to isolate 'a': $$ a = \frac{-55}{6} $$
03

Write the final answer

The solution to the equation is: $$ a = \frac{-55}{6} $$

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Key Concepts

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

Fraction Operations
When solving equations involving fractions, it's essential first to understand how to manage or remove fractions. Fractions can make equations look complex and harder to solve. In the given exercise, we see a fraction \( \frac{6a}{-5} = 11 \). To start solving it, we need to clear the fraction.
  • Multiply both sides of the equation by the denominator of the fraction, in this case, -5. This step helps eliminate the fraction, making the equation easier to handle.
  • In our equation, multiplying both sides by -5 gives us \( (-5) \times \frac{6a}{-5} = 11 \times (-5) \).
  • This operation cancels out the -5 in the fraction, leaving us with a much simpler expression: \( 6a = -55 \).
Clearing fractions is a valuable skill for handling complex equations and will simplify the process of isolating the variable.
Isolate Variable
The essential goal when solving any equation is isolating the variable, such as 'a', to find its value. Once we have cleared fractions from an equation, we focus on getting the variable by itself on one side of the equation.
  • In our current example, after clearing the fraction, we arrived at \( 6a = -55 \).
  • To isolate 'a', divide both sides of the equation by 6, which is the coefficient of 'a'.
  • Doing so gives us \( a = \frac{-55}{6} \), leaving 'a' isolated on one side.
By isolating the variable, we can clearly see the solution for the equation, which helps verify or check our work easily later on. Remembering to perform the same operation on both sides of the equation ensures the balance and equality of the equation is maintained.
Equation Simplification
Equation simplification involves reducing equations to their simplest form. This step can happen either during the process of solving or after having found the solution to ensure it is as clear as possible.
  • When we initially multiply \( (-5) \times \frac{6a}{-5} = 11 \times (-5) \), we simplify by canceling out the fraction's denominator.
  • The resulting equation, \( 6a = -55 \), is already in a simpler form, making it ready to solve by isolating the variable 'a'.
  • Even after finding the solution \( a = \frac{-55}{6} \), checking whether further simplification is possible helps. For instance, checking if the fraction can be reduced further.
Simplification ensures the equation is as straightforward as possible at every step, making it easier to understand and reducing the likelihood of errors.

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Most popular questions from this chapter

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