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Chapter 2: Problem 18

Solve and check. Label any contradictions or identities. $$9-t=21$$

Short Answer

Expert verified
The solution is \(t = -12\).

Step by step solution

01

- Isolate the variable term

Starting with the equation \[9 - t = 21\]simplify by isolating the variable term. Subtract 9 from both sides:\[9 - t - 9 = 21 - 9\]This simplifies to: \[-t = 12\]
02

- Solve for the variable

Now we need to solve for \(t\). Multiply both sides by -1 to isolate \(t\):\[-1(-t) = -1(12)\]This simplifies to: \[t = -12\]
03

- Check the solution

To verify the solution, substitute \(t = -12\) back into the original equation:\[9 - (-12)\]This simplifies to: \[9 + 12 = 21\]Since both sides of the equation are equal, the solution is correct.

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Key Concepts

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

Isolating the Variable
In a linear equation, isolating the variable is essential as it simplifies the equation to a form where the variable is alone on one side. This means we鈥檙e manipulating the equation so that the variable we are solving for is by itself.
In the equation 9 - t = 21, our goal is to get t by itself on one side. To do this, we follow these steps:
  • First, subtract 9 from both sides: 9 - 9 - t = 21 - 9,li>This simplifies to: -t = 12
We use subtraction here to cancel out the constant term (9) that is with the variable. Now, the variable term -t is alone on the left side. This is the first step to solving for t.
Simplifying Equations
Simplifying the equation is about making it easier to handle. Once the variable term is isolated, we often need to perform additional steps to simplify further. Here, after isolating -t,we want to convert -tint_o a positive t .
  • We multiply both sides by -1: -1 * (-t) = -1 * 12
This simplifies to: t = -12Multiplying by -1 effectively changes the sign of our variable term, making the equation simpler to understand and solve. Ensuring the variable term (t ) is positive also makes it clearer to interpret the solution.
Checking Solutions
After solving the equation, it is crucial to verify that our solution is correct by checking it. This step ensures that no mistakes were made during the solving process.
We return to the original equation to substitute the found solution back in:
  • Start with 9 - t = 21
  • Plug in t = -12
  • 9 - (-12)
  • This simplifies to 9 + 12 = 21
Both sides are equal ( 21 = 21), confirming our solution t = -12 is correct. This checking step helps verify the accuracy and builds confidence in the solution.

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