Problem 23 - Prealgebra

Question

Solve each equation. Check your solution. $$3+\frac{t}{2}=35$$

Step-by-Step Solution

Answer

The solution is \( t = 64 \).

Step 1: Isolate the Term with the Variable

Start by isolating the term with the variable. Subtract 3 from both sides of the equation to eliminate the constant term from the left side.\[3 + \frac{t}{2} - 3 = 35 - 3\]This simplifies to:\[\frac{t}{2} = 32\]

Step 2: Solve for the Variable

To solve for \( t \), you'll need to get rid of the fraction. Multiply both sides by 2 to cancel the division by 2.\[2 \times \frac{t}{2} = 32 \times 2\]This simplifies to:\[t = 64\]

Step 3: Check the Solution

Plug the solution back into the original equation to verify it satisfies the equation.Original equation:\[3 + \frac{t}{2} = 35\]Substitute \( t = 64 \):\[3 + \frac{64}{2} = 35\]Simplify the expression:\[3 + 32 = 35\]This is true, so \( t = 64 \) is correct.

Key Concepts

Isolating the Variable

At the heart of equation solving is the concept of isolating the variable. Think of this like peeling an onion to get to its core. You want the variable on one side of the equation, and all the numbers on the other. This helps in unraveling the mystery of what the variable equals.

In our example, we have the equation \( 3 + \frac{t}{2} = 35 \). The goal is to have "\( t \)" all by itself. To start, look at anything added, subtracted, multiplied, or divided with the variable. Here, "3" is added to "\( \frac{t}{2} \)".

To isolate it, subtract 3 from both sides:

• Original equation: \( 3 + \frac{t}{2} = 35 \)
• Subtract 3: \( 3 + \frac{t}{2} - 3 = 35 - 3 \)

After performing the subtraction, you're left with \( \frac{t}{2} = 32 \). Now, "\( t \)" is closer to being isolated, making it easier to solve for.

Step by Step Solution

Breaking down the steps in solving an equation helps in understanding the mechanics behind it. This approach not only aids in finding the solution but also strengthens your grasp of important mathematical operations.

We've already isolated the term with "\( t \)" to be \( \frac{t}{2} = 32 \). Now we must remove any fractions to make the next steps simpler. Notice that "\( t \)" is divided by 2. To cancel that division, multiply both sides of the equation by 2.

Here's how it looks:

• Equation: \( 2 \times \frac{t}{2} = 32 \times 2 \)
• This simplifies to: \( t = 64 \)

By following these steps, we've transformed the process into manageable pieces, resulting in a solution, "\( t = 64 \)". Each action performed kept the equation balanced, ensuring accuracy.

Checking Solutions

Whenever you solve an equation, it's important to verify that the solution is accurate. Ensuring your answer works with the original equation confirms that no mistakes were made during calculations.

To do this, simply substitute your answer back into the original equation. For our example, the original equation was \( 3 + \frac{t}{2} = 35 \).

Let's check it:

• We found \( t = 64 \)
• Substitute 64 into the equation: \( 3 + \frac{64}{2} = 35 \)
• Simplify: \( 3 + 32 = 35 \)

The equality \( 35 = 35 \) holds true, confirming that our solution is indeed correct. Ensuring the correct solution isn't just about finding the right answer—it's about developing the skill to verify the path taken to reach it.