Problem 54 - Prealgebra

Question

Solve each equation. $$5 a-6=14$$

Step-by-Step Solution

Answer

The solution is \(a = 4\).

Step 1: Isolate the variable term

To solve the equation \(5a - 6 = 14\), we need to isolate the term with the variable, which is \(5a\). We start by adding 6 to both sides of the equation to eliminate the constant on the left side. This gives us: \(5a - 6 + 6 = 14 + 6\), simplifying to \(5a = 20\).

Step 2: Solve for the variable

Now, we need to solve for \(a\) by getting rid of the coefficient next to \(a\), which is 5. We do this by dividing both sides of the equation by 5: \(\frac{5a}{5} = \frac{20}{5}\). Simplifying, we find \(a = 4\).

Key Concepts

Variable Isolation

Variable isolation is a crucial step in solving linear equations, like the given example of solving for \( a \) in the equation \( 5a - 6 = 14 \). The idea here is to "free" the variable on one side of the equation. This makes it easier to identify its value.
To begin isolating a variable, such as \( a \), first look to remove any constants involved in its expression. In \( 5a - 6 = 14 \), the \(-6\) is a constant term that we must eliminate to isolate \( 5a \).

- **Step 1:** Add the opposite of this constant, which is +6 in this case, to both sides of the equation. - **Outcome:** This balances the equation and removes the constant from the side with the variable, making it \( 5a = 20 \).

By following this technique, students can methodically consolidate the variable on one side, making subsequent solving steps manageable.

Equation Simplification

Simplifying an equation is an essential part of solving it efficiently. Typically, after isolating the variable term, we simplify the resulting side to make the equation cleaner and more straightforward.
Take the equation after isolating the variable: \( 5a = 20 \). Here, the next task is to simplify this expression such that \( a \) is by itself.

- **Step 1:** Identify the coefficient of the variable; for \( a \), it is 5. - **Step 2:** Divide both sides by this coefficient to remove it from the left side of the equation. - **Calculation:** When you divide \( 5a \) by 5, it leaves you with \( a \) on its own. - **Result:** Performing the division on the other side, \( \frac{20}{5} \), simplifies to 4, so \( a = 4 \).

Simplifying ensures that every transformation applied keeps the equation balanced, leading to the correct solution.

Prealgebra Concepts

Prealgebra concepts provide the foundation for tackling more complex algebraic problems. Understanding these concepts thoroughly is critical for success in higher math.
Let's revisit our original problem of \( 5a - 6 = 14 \). Solving such equations revolves largely around understanding: **variables**, **constants**, and the **properties of equality**.

- **Variables and Constants:** Variables (such as \( a \)) represent unknown values, while constants (like -6 and 14) are fixed numbers.
- **Properties of Equality:** Knowing that you can add, subtract, multiply, or divide both sides of an equation by the same non-zero number is key. This property is at the heart of balancing equations, allowing us to retain equality and find the variable's value.

Mastering these prealgebra concepts ensures that students can approach problems systematically and with confidence.