Problem 8 - Elementary Algebra

Question

Solve each of the equations. $$1.3 t=39$$

Step-by-Step Solution

Answer

The solution is \(t = 30\).

Step 1: Understand the Equation

The equation given is \(1.3t = 39\). You need to find the value of \(t\) that makes this equation true.

Step 2: Isolate the Variable

To isolate \(t\), divide both sides of the equation by 1.3. This will give you \(t = \frac{39}{1.3}\).

Step 3: Perform the Division

Now calculate \(t = \frac{39}{1.3}\). Converting 1.3 into a fraction gives \(\frac{13}{10}\). Inverting it and multiplying by 39 gives us:\[ t = 39 \times \frac{10}{13} \] \[ t = \frac{390}{13} \] \[ t = 30 \]

Step 4: Verify Solution

Substitute \(t = 30\) back into the original equation to ensure it holds true: \(1.3 \times 30 = 39\). The left side equals 39, confirming that the solution is correct.

Key Concepts

Step-by-Step Algebra

Breaking down an equation into manageable steps is key when tackling algebra. Start by understanding the equation you have. Identify what is given and what is unknown. The process usually begins with simplifying each side of the equation, if necessary. Look at each part of the equation carefully. Once you have simplified it as much as possible, move to isolate the variable, which is the unknown you are solving for. This methodical approach ensures you do not miss any details and makes the process easier to follow.

Isolating Variables

Isolating the variable is often the most crucial step in solving linear equations. This involves rearranging the equation so that the unknown variable is on one side by itself. For the equation \(1.3t = 39\), this requires dividing both sides by 1.3. This operation effectively "undoes" the multiplication and makes the variable \(t\) the subject of the equation:\[ t = \frac{39}{1.3} \].

By performing the division, you isolate \(t\) on one side. This maneuver transforms the equation into a simpler form that's easier to solve. Always remember that whatever operation you perform on one side of the equation, you must also perform on the other side. This keeps the equation balanced.

Verifying Solutions

Once you've arrived at a solution, the task isn't over until you've double-checked it. Verification is a way to confirm that your solution is correct. This is done by substituting the found value back into the original equation and ensuring both sides are equal.

For our equation, substituting \(t = 30\), we calculate \(1.3 \times 30 = 39\). Since both sides are equal, we verify that our solution is indeed correct. Verification:

• Plug the solution back into the original equation.
• Calculate both sides of the equation.
• If both sides are equal, the solution is confirmed!

This step catches any errors in calculation and reassures you that your solution is accurate.