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Chapter 2: Problem 30

Solve equation, and check your solution. \(10 s=-54\)

Short Answer

Expert verified
The solution is \[ s = -5.4 \] and it checks out.

Step by step solution

01

Isolate the variable

To isolate the variable, divide both sides of the equation by 10 to solve for s. \[ 10s = -54 \] Divide both sides by 10: \[ s = \frac{-54}{10} \]
02

Simplify the fraction

Simplify the fraction to ensure the solution is in its simplest form. \[ s = \frac{-54}{10} = -5.4 \]
03

Check the solution

Substitute the solution back into the original equation to verify it. Original equation: \[ 10s = -54 \] Substitute \[ s = -5.4 \] into the equation: \[ 10(-5.4) = -54 \] Simplify the left side: \[ -54 = -54 \] Since both sides are equal, the solution is verified.

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Key Concepts

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

Isolate the Variable
The first step in solving a linear equation like this is to isolate the variable. This means getting the variable by itself on one side of the equation.
Let's look at the example equation again: \[ 10s = -54 \] To isolate \( s \), we need to divide both sides of the equation by 10. By dividing both sides by the same number, we maintain the equality. So, \[ \frac{10s}{10} = \frac{-54}{10} \] This simplifies to: \[ s = \frac{-54}{10} \] Now, \( s \) is isolated and we have taken the first step towards solving the equation.
Simplify Fractions
After isolating the variable, the next step is to simplify the fraction we have from step one. The fraction \[ \frac{-54}{10} \] can be simplified by dividing both the numerator and the denominator by their greatest common divisor (GCD).
In our case, both 54 and 10 can be divided by 2. Doing that, we get: \[ \frac{-54 \ \text{divided by} \ 2}{10 \ \text{divided by} \ 2} = \frac{-27}{5} \] Recognizing that -5.4 is the decimal equivalent of \[ \frac{-27}{5} \], we can also write \( s \) as \( -5.4 \). This ensures our solution is in a simpler and more readable form.
Verify Solution
The final step is to verify that our solution is correct. This means substituting the value we found for the variable back into the original equation to ensure both sides are equal.
Our original equation is: \[ 10s = -54 \] We've determined that \( s = -5.4 \). Substitute this value back into the equation: \[ 10(-5.4) = -54 \] Next, multiply on the left side to verify: \[ -54 = -54 \] Since both sides match, our solution \( s = -5.4 \) is verified to be correct. This reassures us that our algebraic manipulations and simplifications lead to the right answer. Always verify your solution to ensure accuracy!

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