Solve for
t
t
. Give your answer in reduced proper fraction or mixed number.

−
3
t
3
+
3
4
=
−
t
−
6
3

Let's solve for \( t \) step-by-step in the following equation:

\[
-3t^3 + \frac{3}{4} = -t - \frac{6}{3}
\]

### Step 1: Simplify the constants
First, simplify the constant on the right side:

\[
\frac{6}{3} = 2
\]

So the equation becomes:

\[
-3t^3 + \frac{3}{4} = -t - 2
\]

### Step 2: Eliminate the fraction
To eliminate the fraction \(\frac{3}{4}\), subtract \(\frac{3}{4}\) from both sides:

\[
-3t^3 = -t - 2 - \frac{3}{4}
\]

### Step 3: Combine the constants
Combine \(-2\) and \(-\frac{3}{4}\):

\[
-2 = \frac{-8}{4}, \quad \text{so} \quad -2 - \frac{3}{4} = \frac{-8}{4} - \frac{3}{4} = \frac{-11}{4}
\]

Now, the equation becomes:

\[
-3t^3 = -t - \frac{11}{4}
\]

### Step 4: Multiply through by 4 to eliminate fractions
Multiply the entire equation by 4 to get rid of the denominator:

\[
4(-3t^3) = 4(-t) + 4\left(\frac{-11}{4}\right)
\]

This simplifies to:

\[
-12t^3 = -4t - 11
\]

### Step 5: Rearrange the equation
Now, move all terms to one side to set the equation to zero:

\[
-12t^3 + 4t + 11 = 0
\]

This is a cubic equation. Solving cubic equations exactly requires methods like factoring, numerical methods, or the cubic formula. Given the complexity of this equation, it’s best to use numerical or graphing methods to approximate the value of \( t \).

Would you like me to continue with numerical approximation or find an exact solution using more advanced methods?

Solve for t. Give your answer in reduced proper fraction or mixed number.
-3t^3 + 3/4 = -t - 6/3

Step-by-step guide to solving the cubic equation -3t^3 + 3/4 = -t - 2, including fraction simplification and elimination. The solution involves algebraic manipulation and concepts suitable for students in Grades 10-12.

Math Problem Statement

− 3 t 3 + 3 4

Solution

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