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Chapter 1: Problem 81

Solve each of the given equations for the indicated variable. $$v=v_{0}+a t\( for \)a$$

Short Answer

Expert verified
The equation for \( a \) is \( a = \frac{v - v_0}{t} \).

Step by step solution

01

Identify the Equation

We are given the equation \( v = v_0 + a t \) and need to solve for the variable \( a \). This means we need to isolate \( a \) on one side of the equation.
02

Rearrange the Equation

To isolate \( a \), subtract \( v_0 \) from both sides of the equation: \( v - v_0 = a t \).
03

Solve for a

Now, divide both sides of the equation by \( t \) to solve for \( a \): \( a = \frac{v - v_0}{t} \).

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Key Concepts

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

Variable Isolation
In algebra, isolating a variable is a fundamental skill you'll often need. Think of it as gathering all information related to one variable in one place. For example, when you're given an equation like \( v = v_0 + a t \), and asked to solve for \( a \), your goal is to get \( a \) by itself on one side of the equation. This is like saying, "Show me what just \( a \) looks like on its own, without anything else attached!"
  • Identify which variable you need to isolate.
  • Move all other terms to the opposite side of the equation.
  • Ensure the isolated variable appears only once in the equation.
Practicing this technique helps you solve and understand more complex equations in algebra.
Rearranging Equations
Rearranging equations might seem challenging at first, but it's simply about reordering the components to make the equation work for us. Let's consider the equation \( v = v_0 + a t \) where you need \( a \). Rearranging here means manipulating the equation so that \( a \) becomes the focal point. Here鈥檚 how you can think about rearranging:
  • Perform the same operation on both sides. This keeps the equation balanced.
  • For \( v = v_0 + a t \), start by moving \( v_0 \) to the other side: \( v - v_0 = a t \). Subtracting \( v_0 \) from both sides is a common step in algebra.
After you rearrange, observe how the equation now clearly highlights \( a \). Ultimately, learning to rearrange equations empowers you to solve algebraic equations more flexibly and rapidly.
Solving for a Variable
Solving for a variable is the final step where you find the precise value of the variable you're interested in. Once you have isolated the variable and rearranged the equation, it鈥檚 time to solve it completely.In our example, after rearranging to \( v - v_0 = a t \), the task is to get \( a \) by itself. This involves:
  • Dividing both sides by \( t \), which leaves you with \( a = \frac{v - v_0}{t} \).
  • Check your work by plugging the values back into the equation to ensure it holds true. This verification step assures accuracy.
Solving for a variable commits you to translate mathematical operations into logical steps. It solidifies your understanding of how all parts play a role to derive a solution. Remember, practice is key, and with time, solving for variables becomes second nature.

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Most popular questions from this chapter

Solve the compound inequality. Express your answer in both interval and set notations, and shade the solution on a number line. $$8 x+5 \leq-1\( and \)4 x-2>-1$$

Solve each of the given equations for \(\mathrm{x}\). $$-5-(9 x+4)=8(-7 x-7)$$

Use interval notation to describe the given set. Also, sketch the graph of the set on the real number line. $$\\{x: x \geq-1\( or \)x>6\\}$$

Perform each of the following tasks in Exercises \(1.5 .96-1.5 .99\) i. Write out in words the meaning of the symbols which are written in set- builder notation. ii. Write some of the elements of this set. iii. Draw a real line and plot some of the points that are in this set. $$A=\\{x \in \mathbb{N}: x>10\\}$$

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