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Chapter 11: Problem 41

Solve using the multiplication principle. Don't forget to check! $$ \frac{-x}{5}=10 $$

Short Answer

Expert verified
The solution is x = -50.

Step by step solution

01

Isolate the variable

To solve the equation \(\frac{-x}{5}=10\), start by isolating the variable \(x\). To do this, multiply both sides of the equation by 5, which is the denominator of the fraction. This will cancel out the fraction on the left side of the equation.
02

Perform the multiplication

Multiply both sides by 5: \[5 \times \frac{-x}{5} = 10 \times 5\] This simplifies to: \(-x = 50\).
03

Solve for x

Now, solve for \(x\) by dividing both sides of the equation by -1: \(\frac{-x}{-1} = \frac{50}{-1}\), which simplifies to \(x = -50\).
04

Check your solution

Verify the solution by substituting \(x = -50\) back into the original equation: \(\frac{-(-50)}{5} = 10\). This simplifies to \(\frac{50}{5} = 10\) which is true, so the solution is correct.

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Key Concepts

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

Isolate the Variable
To solve equations effectively, especially those involving fractions, it鈥檚 crucial to isolate the variable. In the given problem, we have the equation \(\frac{-x}{5}=10\). Here, the variable is \(x\). Our goal is to get \(x\) alone on one side of the equation.
For this, we need to eliminate any fractions. Since \(x\) is divided by 5, multiplying both sides of the equation by 5 will cancel out the denominator. Let鈥檚 see how:
\[5 \times \frac{-x}{5} = 10 \times 5\]
What happens here is: The 5's on the left side cancel each other out, leaving us with:
\[-x = 50\]
Now, we have successfully isolated \(x\), simplifying the expression and gearing up for the next step.
Solve Equation
Once the variable is isolated, we need to solve the equation. In our modified equation, \(-x = 50\), \(x\) is still not completely alone. We need to get rid of the negative sign.
The opposite of multiplying by \(-1\) is dividing by \(-1\), so we perform this operation on both sides of the equation:
\[\frac{-x}{-1} = \frac{50}{-1}\]
This simplifies to:
\[x = -50\]
This step is critical because it finalizes the value of our variable, \(x\). Always remember to perform the same operation on both sides of the equation to maintain the balance.
Check Solution
After finding the value of the variable, it鈥檚 important to check whether our solution is correct.
To verify, substitute \(x = -50\) back into the original equation \(\frac{-x}{5}=10\):
\[\frac{-(-50)}{5} = 10\]
Simplify inside the fraction:
\[\frac{50}{5} = 10\]
And finally, confirm that:
\[10 = 10\]
Since both sides of the equation are equal, the solution \(x = -50\) is indeed correct. This step helps ensure that no mistakes were made during the solving process, offering confidence in the accuracy of the solution.

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