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Chapter 2: Problem 15

Solve and check. Label any contradictions or identities. $$-91=9 t+8$$

Short Answer

Expert verified
The solution is \( t = -11 \)

Step by step solution

01

Isolate the variable term

In order to solve for the variable \( t \), the first step is to isolate the term involving \( t \). Subtract \( 8 \) from both sides of the equation: \[ -91 - 8 = 9t + 8 - 8 \] This simplifies to: \[ -99 = 9t \]
02

Solve for the variable

Now, divide both sides of the equation by \( 9 \) to solve for \( t \): \[ \frac{-99}{9} = \frac{9t}{9} \] Simplify the left side of the equation: \[ t = -11 \]
03

Check the solution

To ensure the solution is correct, substitute \( t = -11 \) back into the original equation and verify if both sides are equal: \[ -91 = 9(-11) + 8 \] Calculate the right-hand side: \[ 9(-11) + 8 = -99 + 8 = -91 \] Since \( -91 = -91 \), 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
To solve a linear equation, the first crucial step is to isolate the variable. This means we want to get the variable term by itself on one side of the equation. Starting with our exercise, we have: \( -91 = 9t + 8 \).
To isolate the variable term, we need to eliminate the constant term that's on the same side as the variable. Here, the constant term is 8. We can remove it by subtracting 8 from both sides of the equation: -91 - 8 = 9t + 8 - 8
This simplification leaves us with: -99 = 9t.
By doing this, we've successfully isolated the term involving our variable, making it easier to solve for t in the next step.
Verify Solution
After finding the value of the variable, it鈥檚 important to check if our solution is correct. Verification ensures that there are no mistakes in the calculations.
In our example, our solution gave us t = -11. To verify, we substitute this value back into the original equation: \( -91 = 9(-11) + 8 \) Let's calculate the right-hand side: 9(-11) + 8 = -99 + 8 = -91. Notice how both sides of the equation are equal, confirming our solution.
  • This process helps us ensure our steps and calculations are correct.
  • Always adopt this habit to minimize errors.
By verifying, we鈥檝e assured that t = -11 is indeed the correct solution to the given equation.
Step-by-Step Solution
Breaking down the solution into clear and manageable steps is a great way to understand how we solve linear equations. Each step builds on the previous one, creating a clear path to the solution.
1. **Isolate the variable term:** Start with the equation \( -91=9t+8 \) Subtract 8 from both sides: -99 = 9t 2. **Solve for the variable:** Once isolated, divide both sides by 9 to solve for t: \( \frac{-99}{9} = \frac{9t}{9} \) This simplifies our variable: \( t = -11 \)
3. **Check the solution:** Substitute t = -11 back into the original equation to verify: \( -91 = 9(-11) + 8 \) Calculate the right-hand side: -91 = -99 + 8 = -91
  • By following these steps, students can systematically approach the problem.
  • Understanding each individual step helps in grasping the overall concept.
  • This methodical process ensures accuracy and builds confidence in solving equations.

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