Solve the linear equation:

• (3t - 2)/4 - (2t + 3)/3 = 2/3 - t

To solve this equation, first, find a common denominator for the fractions, which is 12.

• Multiply each term by 12:
• 12 * (3t - 2)/4 - 12 * (2t + 3)/3 = 12 * (2/3) - 12 * t

This simplifies to:

• 3(3t - 2) - 4(2t + 3) = 8 - 12t

Expanding the left side:

• 9t - 6 - 8t - 12 = 8 - 12t

Combine like terms:

• t - 18 = 8 - 12t

Add 12t to both sides:

• t + 12t - 18 = 8

This simplifies to:

• 13t - 18 = 8

Now, add 18 to both sides:

• 13t = 26

Finally, divide by 13:

• t = 2

To solve the linear equation \((3t - 2)/4 - (2t + 3)/3 = 2/3 - t\), we can follow a systematic approach. Let's break it down step by step, ensuring we handle the fractions carefully.

Finding a Common Denominator

The first step involves identifying a common denominator for the fractions present in the equation. The denominators here are 4, 3, and 1 (since \(t\) can be considered as \(t/1\)). The least common multiple of these numbers is 12. This means we will multiply every term in the equation by 12 to eliminate the fractions.

Multiplying Through by 12

Now, let's multiply each term by 12:

• 12 * \((3t - 2)/4\) becomes \(3(3t - 2)\)
• 12 * \((2t + 3)/3\) becomes \(4(2t + 3)\)
• 12 * \((2/3)\) becomes \(8\)
• 12 * \(-t\) becomes \(-12t\)

Putting it all together, we have:

3(3t - 2) - 4(2t + 3) = 8 - 12t

Expanding the Equation

Next, we need to expand the left side of the equation:

• 3(3t - 2) = 9t - 6
• 4(2t + 3) = 8t + 12

Substituting these back into the equation gives us:

9t - 6 - (8t + 12) = 8 - 12t

This simplifies to:

9t - 6 - 8t - 12 = 8 - 12t

Combining Like Terms

Now, let's combine like terms on the left side:

• 9t - 8t = t
• -6 - 12 = -18

This results in:

t - 18 = 8 - 12t

Isolating the Variable

To isolate \(t\), we can add \(12t\) to both sides:

t + 12t - 18 = 8

This simplifies to:

13t - 18 = 8

Solving for t

Next, we add 18 to both sides to further isolate \(t\):

13t = 26

Finally, we divide both sides by 13:

t = 2

Final Result

The solution to the equation \((3t - 2)/4 - (2t + 3)/3 = 2/3 - t\) is \(t = 2\). This process illustrates how to handle linear equations with fractions by finding a common denominator, expanding, and combining like terms effectively.