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Chapter 2: Problem 29

Solve equation, and check your solution. \(10 t=-36\)

Short Answer

Expert verified
The solution is \( t = \frac{-18}{5} \).

Step by step solution

01

- Isolate the variable

To solve for the variable, the objective is to isolate it. This means we need to have the variable alone on one side of the equation. The given equation is: \[ 10t = -36 \] To isolate \( t \), divide both sides of the equation by 10.
02

- Perform the division

Next, perform the division: \[ t = \frac{-36}{10} \] Simplify the fraction to get the value of \( t \).
03

- Simplify the fraction

Now, simplify \( \frac{-36}{10} \) by finding the greatest common divisor (GCD). The GCD of 36 and 10 is 2. So, divide both the numerator and the denominator by 2 to get: \[ t = \frac{-36 \text{ 梅 } 2}{10 \text{ 梅 } 2} = \frac{-18}{5} \] Therefore, \( t = \frac{-18}{5} \).
04

- Check the solution

To verify the solution, substitute \( t = \frac{-18}{5} \) back into the original equation: \[ 10t = -36 \] Substitute \( t \): \[ 10 \times \frac{-18}{5} = -36 \] Calculate: \[ \frac{10 \times -18}{5} = -36 \] \[ \frac{-180}{5} = -36 \] Since \( -36 = -36 \) is a true statement, the solution is verified.

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Key Concepts

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

Isolating Variables
When solving linear equations, isolating the variable is a key step.
This means getting the variable you are solving for by itself on one side of the equation.
In our example, we start with the equation \(10t = -36\).
To isolate \(t\), we need to get rid of the 10 that is multiplying it.
To do this, we divide both sides of the equation by 10:
\[t = \frac{-36}{10}\text{.} \]
This step makes it easier to see what the variable equals.
Once the variable is isolated, we can find its value.
Remember that whatever operation you do on one side of the equation, you must do the same to the other side.
Simplifying Fractions
Simplifying fractions helps make our answers cleaner and more understandable.
In our equation, we end up with \[t = \frac{-36}{10}\text{.}\]
To simplify this fraction, find the greatest common divisor (GCD) of the numerator and the denominator.
The GCD of 36 and 10 is 2.
Now divide both the numerator and the denominator by 2:
\[t = \frac{-36 \text{ 梅 } 2}{10 \text{ 梅 } 2} = \frac{-18}{5}\text{.} \]
Simplifying makes the fraction easier to work with and better for checking the solution.
Simplified fractions are also usually preferred in final answers.
Checking Solutions
After solving for the variable, it鈥檚 essential to check the solution.
Substitute the solution back into the original equation to verify it works.
For our solution, \(t = \frac{-18}{5}\), substitute it back into the original equation:
\[10t = -36\]
Substitute \(t\):
\[10 \times \frac{-18}{5} = -36\]
Perform the multiplication:
\[ \frac{10 \times -18}{5} = -36\]
\[ \frac{-180}{5} = -36\]
Simplify the fraction:
\[ -36 = -36\]
Since both sides of the equation are equal, our solution is verified.
Always check your work to ensure your solution accurately satisfies the original equation.

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