Problem 20 - College Algebra and Calculus: An Applied Approach

Question

Solve the equation and check your solution. (Some equations have no solution.) $$ 7 x+2=16 $$

Step-by-Step Solution

Answer

The solution to the equation \(7x + 2 = 16\) is \(x = 2\).

Step 1: Isolate the Variable

First, we need to isolate the variable(). We start by subtracting \(2\) from both sides of the equation: \(7x + 2 - 2 = 16 - 2\), which simplifies to \(7x = 14\).

Step 2: Solve for the Variable

Next, we have \(7x = 14\). Divide both sides by \(7\) to get \(x\): \(x = 14 / 7\), which further simplifies to \(x = 2\).

Step 3: Verify the Solution

Lastly substitute \(x = 2\) back into the original equation: \(7 * 2 + 2 = 16\). This simplifies to \(14 + 2 = 16\), which is true, confirming that our solution \(x = 2\) is correct. Confirmation of the solution is especially important in case the equation had no solution, which is not the case here.

Key Concepts

Isolate the Variable

Mastering the art of solving linear equations hinges on one crucial step: isolating the variable. This involves manipulating the equation to obtain the variable on one side while keeping the constant terms on the other. It's a bit like tidying a room by putting toys in the toy box and books on the shelf, so you have a clear path to walk—a neat equation makes it easier to see the solution.

Take the example, \(7x + 2 = 16\). Here, we aim to 'clear the path' for \(x\) by eliminating the constant term on the same side as \(x\). We do this by performing the inverse operation to both sides - in this case, subtracting \(2\) from each. This balances the equation, maintaining its equality, while isolating the variable, as seen in the first step of the textbook solution.

Why Is Isolation Important?

Isolating the variable is a foundational technique because it transforms the equation into a simpler form that can be readily solved. Without isolation, deducing the value of the variable might remain an elusive or overly complex task.

Equation Solving Steps

Solving equations can be broken down into methodical steps to make the process less daunting. The key to conquering any equation is consistency and systematic work. Our equation, \(7x+2=16\), serves as an excellent example to demonstrate these steps.

First, as previously mentioned, isolate the variable. Once the variable \(x\) stands alone, you can proceed to the second step: resolving the equation. In our given example, once isolated, you are left with \(7x = 14\). To solve for \(x\), you need to 'undo' the multiplication by 7, which you achieve by dividing both sides by 7, leading to \(x = 2\). This step essentially reverses the operations that have been applied to the variable.

Tips for Smooth Solving

• Perform operations on both sides equally to maintain the balance.
• Work in the reverse order of operations - 'undoing' the equation.
• Remain calm and methodical, even if the equation becomes more complex.

Following these guidelines will lead you to the correct solution while also building a solid understanding of the problem-solving process.

Verifying Algebraic Solutions

Solving the equation is just part of the journey; verification is your confirmation ticket. Verifying ensures that your proposed solution truly satisfies the original equation. It's like checking your shopping list when you get home to make sure you didn't forget anything important. In our example, after finding that \(x = 2\), we substitute it back into the original equation: \(7 * 2 + 2 = 16\). This confirms our solution as correct, as it simplifies to \(14 + 2 = 16\), which holds true.

Why Verification Matters

Verification protects against common errors, such as arithmetic mistakes or incorrect manipulation of terms. It is an essential skill, particularly in more complex equations where the likelihood of mistakes increases. It also reinforces learning by highlighting the relationship between each step of the equation solving process. In educational settings, proper verification earns you full marks and in real-world applications, it ensures your solutions are reliable. This step is a non-negotiable finale to the problem-solving ballet, ensuring that your hard work yields a correct and useful result.