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Chapter 5: Problem 100

Solve each equation. $$ 3 t+2=0 $$

Short Answer

Expert verified
t = -\frac{2}{3}

Step by step solution

01

Isolate the Variable

To solve for the variable, isolate it on one side of the equation. Start with the equation: \[ 3t + 2 = 0 \] Subtract 2 from both sides to get: \[ 3t = -2 \]
02

Solve for the Variable

Divide both sides of the equation by 3 to isolate the variable t: \[ t = -\frac{2}{3} \]
03

Check the Solution

Substitute \( t = -\frac{2}{3} \) back into the original equation to ensure it satisfies the equation: \[ 3(-\frac{2}{3}) + 2 = 0 \] Simplifies to: \[ -2 + 2 = 0 \] Which is true. Therefore, the solution is correct.

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Key Concepts

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

Isolate the Variable
When solving linear equations, the first step is to isolate the variable. This means we need to get the variable alone on one side of the equation.
Consider the given equation: \( 3t + 2 = 0 \).

To isolate \( t \), we must start by removing the constant term on the same side as the variable. In this case, the constant is 2. We do this by subtracting 2 from both sides of the equation:
\[ 3t + 2 - 2 = 0 - 2 \]
Which simplifies to:
\[ 3t = -2 \]
Now, the variable term \( 3t \) is isolated on one side of the equation, and we can proceed to the next step. The goal is to work step-by-step to ensure accuracy.
Solve for the Variable
After isolating the variable, the next step is to solve for it. This involves getting the variable on its own by performing any necessary arithmetic operations.
We have the simplified equation from the earlier steps:
\[ 3t = -2 \]

To solve for \( t \), we need to eliminate the coefficient of 3. This is done by dividing both sides by 3:
\[ t = \frac{-2}{3} \]

This gives the value of the variable \( t \). It鈥檚 important to perform the same operation on both sides of the equation to maintain equality. Now, we have found that \( t = -\frac{2}{3} \). Always ensure each step follows the basic principles of algebra.
Check the Solution
Finally, it鈥檚 crucial to check the solution to ensure it鈥檚 correct.
We found that \( t = -\frac{2}{3} \). To verify, substitute this value back into the original equation:
\[ 3 \bigg( -\frac{2}{3} \bigg) + 2 = 0 \]

First, multiply the terms:
\[ -2 + 2 = 0 \]

Since both sides of the equation equal 0, the solution is verified to be correct.

Always remember to check your work to catch any possible mistakes. Double-checking can make a big difference and helps ensure understanding.

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Most popular questions from this chapter

Factor completely. \(v^{2}-11 v w+30 w^{2}\)

A student factored \(x^{3}+4 x^{2}-2 x-8\) as follows. $$\begin{aligned} x^{3} &+4 x^{2}-2 x-8 \\ &=\left(x^{3}+4 x^{2}\right)+(-2 x-8) \\ &=x^{2}(x+4)+2(-x-4) \end{aligned}$$ The student could not find a common factor of the two terms. WHAT WENT WRONG? Complete the factoring.

Solve each equation, and check your solutions. $$6 p^{2}(p+1)=4(p+1)-5 p(p+1)$$

Factor each trinomial completely. $$ 36 y^{2}-60 y+25 $$

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