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Chapter 3: Problem 111

Is 8 a solution of the equation \(\frac{m}{-4}=\frac{-16}{m} ?\)

Short Answer

Expert verified
Yes, 8 is a solution to the given equation.

Step by step solution

01

Substitute the value of m

Replace m by 8 in the given equation. After substitution, the equation will be: \(\frac{8}{-4}=\frac{-16}{8}\).
02

Simplify

Simplify both sides of the equation. In the left side, \(8/-4=-2\) and on the right side \(-16/8 = -2 \). So, the simplified equation will be -2=-2.
03

Verify

Since after substitution and simplification, both sides of the equation equal each other, therefore we can say that 8 is indeed a solution to the given equation.

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Key Concepts

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

Equation Solving
The process of solving an equation involves finding the value of the variable that makes the equation true.
In simple terms, we want to find the number that can replace the variable in the equation to make both sides equal.
In the given exercise, the equation is \( \frac{m}{-4}=\frac{-16}{m} \).
To check if 8 is a solution, we need to determine if substituting 8 for \( m \) satisfies the equation.
Substitution Method
The substitution method is a straightforward and commonly used technique in equation solving.
It involves replacing the variable in an equation with a given number to see if the equation holds true.
For the problem at hand, we're checking whether 8 is a solution.
  • Substitute 8 in place of \( m \) in the equation: \( \frac{8}{-4} = \frac{-16}{8} \).
  • By doing this, we transform the equation into a numerical form, allowing us to work with plain numbers.
By substituting, we essentially test if our guess (in this case, 8) is correct.
Simplification
Once substitution is complete, the next step is simplification.
Simplification means reducing the equation to its simplest form by performing any arithmetic operations required.
In our example:
  • On the left side: \( \frac{8}{-4} = -2 \).
  • On the right side: \( \frac{-16}{8} = -2 \).
Both sides simplify to the same value, \(-2\), which is an important indicator during equation solving.
When both sides match after simplification, it suggests our substituted value might be correct.
Verification Process
Verification is the final and crucial step in solving equations.
It involves confirming that the simplified equation holds true for the value we substituted.
In this case, after substitution and simplification, both sides of the equation equate to \(-2\).
Since they are equal, we verify that our substitution was correct.
This confirms that 8 is indeed a solution to the equation \( \frac{m}{-4}=\frac{-16}{m} \).
  • This step ensures that no arithmetic or substitution errors have occurred.
  • It reassures that our solution is valid and we successfully solved the equation.
Verification acts as a completeness check for your solution.

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Most popular questions from this chapter

Simplify. $$5+4-(-3)-7$$

Evaluate the expression for the given values of the variables. \(x-(-y),\) for \(x=-3\) and \(y=9\)

Is \(-8\) a solution of the equation \(6=-3+z ?\)

Make up three subtraction problems such that each problem involves a negative number minus a negative number, and each problem has a difference of \(-8 .\) Then describe a strategy for writing these problems.

Is \(-8\) a solution of the equation \(-7+m=-15 ?\)
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