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Chapter 5: Problem 26

Solve the equations. $$-4.1 w=-73.8$$

Short Answer

Expert verified
w = 18

Step by step solution

01

- Isolate the variable

To solve for the variable \( w \), divide both sides of the equation by \(-4.1\). The equation is given as \[ -4.1w = -73.8 \] Dividing both sides by \(-4.1\), we get: \[ w = \frac{-73.8}{-4.1} \]
02

- Simplify the division

Simplify the fraction \(\frac{-73.8}{-4.1}\). The negative signs will cancel each other out, and we need to divide 73.8 by 4.1: \[ w = 18 \]
03

- Verify the solution

To ensure our solution is correct, substitute \( w = 18 \) back into the original equation: \[ -4.1 \times 18 = -73.8 \] This confirms the solution is correct since both sides are equal.

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Key Concepts

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

Isolate the variable
When solving linear equations, the goal is to isolate the variable on one side of the equation. This means we want the variable by itself. In this exercise, the variable is \(w\). The equation we start with is \[ -4.1w = -73.8 \].
To isolate \(w\), we need to get rid of the \(-4.1\) that is multiplied by \(w\).
We do this by dividing both sides of the equation by \(-4.1\). Remember, whatever you do to one side of the equation, you must do to the other to keep it equal.
So, \[ \frac{-4.1w}{-4.1} = \frac{-73.8}{-4.1} \]
Now, we have \(w\) isolated: \[ w = \frac{-73.8}{-4.1} \].
Simplifying fractions
Once we have isolated the variable and have an expression like \[ w = \frac{-73.8}{-4.1} \], the next step is to simplify the fraction. When simplifying fractions, the idea is to make the fraction as simple as possible.
In this case, observe the negative signs in both the numerator and the denominator. Negative divided by negative gives a positive result. Therefore, the negative signs cancel each other out: \[ w = \frac{73.8}{4.1} \].
Next, perform the division: 73.8 divided by 4.1 equals 18. This gives us our simplified solution: \[ w = 18 \].
Verify solution
Verify that our solution is correct by substituting it back into the original equation. Verifying helps us catch any mistakes we might have made.
Our original equation is \[ -4.1w = -73.8 \].
Substitute \(w = 18\) back into the equation: \[ -4.1 \times 18 \].
Calculate the right-hand side: \[ -4.1 \times 18 = -73.8 \].
Since both sides equal \(-73.8\), we confirmed that our solution \( w = 18 \) is correct. Always substitute the solution back into the original equation to ensure it works.

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