Problem 41 - Elementary Algebra

Question

Solve. $$ 3 a-7=23 $$

Step-by-Step Solution

Answer

The solution to the equation is \(a = 10\).

Step 1: Add 7 to Both Sides

The first step in isolating the variable is to eliminate the constant on the left side of the equation. We achieve this by adding 7 to both sides of the equation. This can be written as: \[3a - 7 + 7 = 23 + 7\]This simplifies to:\[3a = 30\]

Step 2: Divide Both Sides by 3

The next step is to solve for \(a\). Since \(a\) is multiplied by 3, we need to divide both sides of the equation by 3 to isolate \(a\). This can be shown as:\[\frac{3a}{3} = \frac{30}{3}\]This results in:\[a = 10\]

Step 3: Solution Verification

To ensure the solution is correct, substitute \(a = 10\) back into the original equation to verify:\[3(10) - 7 = 23\]This simplifies to:\[30 - 7 = 23\]Since both sides equal 23, the solution \(a = 10\) is correct.

Key Concepts

Isolation of Variables

Isolation of variables is a fundamental concept in solving linear equations. The goal is to manipulate the equation to get the unknown variable by itself on one side. In our example, the given equation is \(3a - 7 = 23\). To isolate the variable \(a\), we first need to eliminate the constant \(-7\) from the left side.

One effective method is to perform the opposite operation. Since \(-7\) is being subtracted, we add \(7\) to both sides of the equation. The principle here is whatever you do to one side of the equation, you must do to the other to maintain balance. This leads us to a simpler equation: \(3a = 30\). Now, the variable is closer to being isolated.

Finally, we divide by \(3\) to solve for \(a\), resulting in \(a = 10\). The variable \(a\) is now isolated, offering us its value.

Algebraic Manipulation

Algebraic manipulation involves performing operations to both sides of an equation to alter it into a more useful form. It's the art of simplifying equations while keeping them balanced.

Let's consider our original equation: \(3a - 7 = 23\). By understanding that \(-7\) counteracts with its positive form, we added \(7\) to both sides. This neutralizes \(-7\) and leaves us with a simpler equation: \(3a = 30\).

Next, since \(3\) is multiplied by \(a\), the inverse operation—division—is used. Dividing both sides by \(3\) helps cancel the three, resulting in \(a = 10\).

The same operations performed on both sides ensure the equation remains balanced. Good algebraic manipulation keeps the integrity of equations without introducing errors.

Solution Verification

Solution verification is an essential step to ensure that the answer obtained is correct. It involves substituting the solution back into the original equation to see if the equation holds true.

For our solution, \(a = 10\), we plug it back into the initial equation: \(3(10) - 7 = 23\). Simplifying the left side gives \(30 - 7\), which equals \(23\).

Since both sides of the equation are equal, we can be confident that \(a = 10\) is the correct solution. Verification helps catch any errors made in the process of solving the equation and reinforces the correct application of algebraic methods.

Always verify your solutions to confirm correctness and gain assurance of your mathematical process.