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Chapter 2: Problem 35

Solve each equation. $$-t+3=27$$

Short Answer

Expert verified
The solution to the equation is \( t = -24 \).

Step by step solution

01

Isolate the variable

Start by isolating the variable \( t \) on one side of the equation. To do this, subtract 3 from both sides of the equation:\[-t + 3 - 3 = 27 - 3\]This simplifies to:\[-t = 24\]
02

Solve for the variable

Now, you need to find the value of \( t \). Since \(-t = 24\), multiply both sides by \(-1\) to solve for \( t \):\[-1(-t) = -1(24)\]This simplifies to:\[t = -24\]

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Key Concepts

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

Isolating Variables
In a linear equation, isolating the variable is your first step. The goal here is to have the variable by itself on one side of the equation. Begin with the original equation:
  • The equation was \(-t + 3 = 27\)
  • To isolate \(t\), you first need to get rid of anything that is added or subtracted from it.
  • In this case, you have \(+3\) on the same side as \(-t\).
So, you subtract 3 from both sides to maintain the equation's balance:\(-t + 3 - 3 = 27 - 3\).
That's how you are left with \(-t = 24\).
Now the variable is isolated, as \(-t\) is the only thing on the left side.
Solving for a Variable
After isolating the variable, the next goal is to solve for it. What does this mean? It means finding the actual value of the variable that makes the equation true.
In our exercise, once you have \(-t = 24\), the task is to find the actual value of \(t\).
  • Multiply through by \(-1\), as \(-1\) times \(-t\) will give you \(t\).
  • Apply this operation on both sides: \(-1(-t) = -1(24)\).
  • This operation is vital because you want the variable \(t\) alone, without any negative coefficient.
The equation then becomes \(t = -24\).
This means \(-24\) is the value of \(t\) that makes the original equation true.
Negative Coefficients
In linear equations, dealing with negative coefficients requires careful handling. A negative coefficient indicates that the variable is multiplied by \(-1\).
In this specific problem, the term for \(t\) was \(-t\), implying a negative coefficient.
To simplify things and solve the equation:
  • We multiply both sides of the equation by \(-1\) to change \(-t\) into a positive \(t\).
  • This is an essential step, anytime you are faced with a negative in front of your variable.
  • The result of this operation yields: \(t = -24\).
Remember, resolving negative coefficients correctly is crucial to finding the accurate solution to any equation.
It's all about turning that variable positive while maintaining the balance of the equation.

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