Problem 36 - Prealgebra

Question

Using the addition property of equality first, solve each of the following equations. $$-12 a+1=-47$$

Step-by-Step Solution

Answer

The solution is \(a = 4\).

Step 1: Eliminate the Constant Term

Start with the given equation: \(-12a + 1 = -47\).The goal is to isolate the variable term on one side. First, eliminate the constant term on the left side by subtracting 1 from both sides of the equation:\(-12a + 1 - 1 = -47 - 1\).This simplifies to:\(-12a = -48\).

Step 2: Solve for the Variable

Now that we have \(-12a = -48\),our goal is to solve for \(a\). To do this, divide both sides of the equation by -12:\(\frac{-12a}{-12} = \frac{-48}{-12}\).This simplifies to:\(a = 4\).

Key Concepts

Solving Equations

Solving equations is a fundamental skill in mathematics, where you aim to find the value of the unknown variable that makes an equation true. Think of it like a balance scale, where both sides must remain equal.
When solving an equation, the steps you take must ensure that balance is maintained. Different equations might require different methods, but the underlying idea is consistently to simplify and break down until you isolate the unknown variable.
For example, in the equation \[-12a + 1 = -47\]Our task is to find the value of \(a\). We start by simplifying each side of the equation.
Keep in mind that every operation performed on one side must be performed on the other side too. This principle ensures that the equation remains balanced and true.

Isolation of Variables

The isolation of variables is the process of rearranging an equation to get the unknown variable on its own, typically on one side of the equation. This is critical because having the variable by itself helps to easily see the solution.
Using the addition property of equality is a powerful technique in achieving this. It involves adding or subtracting the same amount from both sides of the equation to cancel out terms. For our equation:\(-12a + 1 = -47\)1. We subtract 1 from both sides to remove the constant term on the left: \(-12a + 1 - 1 = -47 - 1\), simplifies to \(-12a = -48\)
2. With the constant removed, focus turns to the variable negative coefficient, which in this case is -12.
By understanding and applying these steps, isolation becomes straightforward.

Mathematical Operations

Mathematical operations are the backbone of solving equations, emphasizing addition, subtraction, multiplication, and division to manipulate expressions. These operations must be applied carefully to ensure the integrity of the equation remains intact.
In our example:\(-12a = -48\)We need \(a\) alone. Since \(a\) is being multiplied by -12, we use the opposite operation—division. Here’s how:- Divide each side by -12 to counteract the multiplication: \(\frac{-12a}{-12} = \frac{-48}{-12}\)
- This results in \(a = 4\), which is the solution.
Mastering these operations allows one to handle equations efficiently, aiding in quick and accurate solutions through well-understood processes.