Problem 30 - Algebra 1: Concepts and Skills

Question

Solve the equation. \(9 t-15 t=-18\)

Step-by-Step Solution

Answer

The solution to the equation is \(t = 3\).

Step 1: Simplify the Equation

Combine like terms on the left side of the equation. In this case, combine the \(9t\) and the \(-15t\) to get a simpler equation: \(-6t = -18\).

Step 2: Solve the Equation

Now it is need to isolate t, by dividing both sides of the equation by -6: \(t = -18 / -6 = 3\).

Key Concepts

Combining Like Terms

When dealing with algebra equations, one of the first steps is often to combine like terms. "Like terms" are terms that have the same variables raised to the same powers. In simpler terms, they are similar because of their variable components.

Let's take a look at the example from our exercise:

• We started with the expression on the left side: \(9t - 15t\).
• Both terms include the variable \(t\), making them like terms.
• When we combine them, we merge their coefficients. This means we perform the operation \(9 - 15\), which gives us \(-6t\).

This step is crucial as it simplifies the equation from its initial state into something easier to work with. Combining like terms transforms equations by reducing the number of terms, paving the way for easier solutions.

Solving Linear Equations

Once we combine like terms, our equation becomes simpler, yet there's still work to be done. Solving linear equations involves finding the values of the variables that make the equation true. In our case:

After combining like terms, the equation was transformed to:

• \(-6t = -18\)

We classify this as a linear equation because it has variables raised only to the first power. Linear equations typically look like \(ax + b = c\), where \(a\), \(b\), and \(c\) are constants.

The straightforward aspect of solving linear equations is their predictability. We follow a structured process, which often involves isolating the variable, our next core concept.

Isolating Variables

To solve any equation efficiently, especially linear ones, we often need to isolate the variable of interest. This means manipulating the equation until the variable stands alone on one side. In our example:

Once we reached \(-6t = -18\), we aimed to isolate \(t\).

• The operation we performed was dividing both sides by \(-6\).
• This resulted in \(t = 3\).

Why divide by \(-6\)? Because it's the coefficient of \(t\), and dividing both sides by this number cancels it out on the left side. It leaves \(t\) by itself, achieving our goal of isolation.

Isolating variables is a key step and is necessary to uncover the solutions of equations. It encapsulates the elegance of algebra: transforming equations until answers reveal themselves.