Problem 11 - Algebra 1: Concepts and Skills

Question

Solve the equation. \(13=12 t-5-3 t\)

Step-by-Step Solution

Answer

The solution to the equation \(13=12t-5-3t\) is \(t = 2\).

Step 1: Combine Like Terms

First, combine the terms that contain variable 't'. This will result in the equation: \(13 = 12t - 3t - 5\), which simplifies to: \(13 = 9t - 5\).

Step 2: Isolate the Variable

Second, start isolating the variable 't'. To do this, add 5 to both sides of the equation to move '-5' from the right side to the left. This results in: \(13 + 5 = 9t - 5 + 5\), which simplifies to: \(18 = 9t\).

Step 3: Solve for t

Lastly, divide both sides of the equation by 9 to solve for 't'. This gives: \(18 / 9 = 9t / 9\), which simplifies to: \(t = 2\).

Key Concepts

Combining Like Terms

When solving linear equations, one of the first steps you should take is to combine like terms. This means adding or subtracting terms that have the same variable raised to the same power. In our exercise, we encounter terms with the variable 't': 12t and -3t. These can be combined because they both represent multiples of 't'.

To combine them, simply add or subtract their coefficients (the numbers in front of the variable). Here, we subtract the coefficients: 12 - 3 = 9, which leads to the term 9t. This simplification results in a cleaner equation and makes it easier to isolate the variable in the next steps. Remember, this step only applies to terms that are similar—terms that have the same variable part. Constants or terms with different variables are not combined in this way.

Isolating the Variable

Isolating the variable is a crucial step in solving linear equations. It involves moving all terms with the variable onto one side of the equation and all constants to the other side. Our goal is to get the variable 't' by itself. In the given exercise, we begin with 13 = 9t - 5.

We need to move the '-5' to the other side of the equation to isolate 't'. To do this, we perform the inverse operation, which in this case means adding 5 to both sides, resulting in 13 + 5 = 9t. Now the equation is down to a single term with the variable 't' and a constant on the other side. This is the setup that we aim for before moving on to the last step, which is actually solving for the variable.

Solving for a Variable

The last step in solving linear equations is to solve for the variable. This usually means performing an operation that will leave the variable by itself on one side of the equation. At this point in our exercise, we have simplified the equation to 18 = 9t.

To get 't' by itself, we divide both sides of the equation by the coefficient of 't', which is 9. Dividing both sides by the same number ensures that the balance of the equation is maintained. Thus, 18/9 = 9t/9 simplifies to t = 2, which is our final solution. It's essential to perform the same operation on both sides of the equation to keep it equal—that is, to satisfy the original condition set by the equation.