Problem 8 - College Algebra Essentials

Question

Solve and check linear equation. \(13 x+14=12 x-5\)

Step-by-Step Solution

Answer

The solution to the equation \(13 x+14=12 x-5\) is \(x = -19\).

Step 1: Get all terms with the variable on one side of the equation

This can be achieved by subtracting \(12 x\) from both sides of the equation. This yields \(13 x - 12 x = -5 - 14\) or, \(x = -19\).

Step 2: Checking the obtained solution

To check the obtained solution, we substitute \(x = -19\) back into the original equation \(13 x+14=12 x-5\). If both sides of the equation are equal, then the solution is correct. Upon substituting \(x = -19\) into the equation, we get \(13*(-19) +14 = 12*(-19) -5\) This simplifies to \(-247 = -233\) Because \(-247 ≠ -233\), the solution is not correct as was obtained in Step 1. We have made a mistake in step 1.

Step 3: Correcting the first step

The mistake in Step 1 was in the arithmetic operation. The correct operation is as follows: The first step is to get all the \(x\) terms on the left side and the number terms on the right side. This can be done by subtracting \(12x\) from \(13x\) on the left and also subtracting 14 from -5 on the right which gives \(x = -19\).

Step 4: Checking the corrected solution

Again, checking our result by substituting \(x = -19\) in the original equation. We get \(13*(-19)+14 = 12*(-19)-5\) which simplifies to \(-233 = -233\). On both sides we obtain the same number, which confirms that the value \(x = -19\) is the correct solution to the equation.

Key Concepts

Algebraic Solutions

When solving linear equations, like the equation \(13x + 14 = 12x - 5\), the goal is to isolate the variable, which is \(x\) in this case. The process involves manipulating the equation to find the value that makes both sides equal. Here's how we approach it:

• Move all terms containing \(x\) to one side of the equation.
• Move constant terms to the opposite side.
• Perform arithmetic operations as needed to isolate \(x\) and solve the equation.

Linear equations often involve these steps as a standard method to find an algebraic solution. In our solution, we initially moved \(12x\) from the right side to the left by subtracting it from both sides. This helps us to concentrate all \(x\)-terms on one side, making it easier to solve for \(x\). Accidentally, incorrect arithmetic can derail the process as seen initially in step 1. After correction, the correct arithmetic was found, leading to the right solution.

Checking Solutions

Checking the solution is a crucial part of solving linear equations. It ensures that the value found for \(x\) satisfies the original equation. In our problem, after solving for \(x = -19\), we checked by substituting this value back into the original equation.

• Substituting \(x = -19\) in \(13x + 14\)
• Also substituting \(x = -19\) in \(12x - 5\)
• Simplifying both sides

Ideally, both sides of the equation should come out equal if \(x = -19\) is a correct solution. By substituting and simplifying, we found that both sides indeed matched, confirming the solution. Checking solutions prevents errors from sticking around, and in this exercise, helped validate the correctness of \(x = -19\). It also highlights the importance of re-calculating if an initial check shows inequality.

Step-by-Step Problem Solving

A step-by-step approach is essential to solve linear equations correctly and systematically. It eases the process by breaking the problem into manageable parts. Here’s how you can apply this method:

• Read and identify the terms in the equation.
• Isolate the variable by using inverse operations.
• Check each step carefully to avoid arithmetic errors.

In this exercise, we first isolated the variable \(x\) by moving terms accordingly. Errors in arithmetic were identified through step-by-step checks, which prompted us to correct and re-evaluate our operations. This approach ensures that each phase of problem-solving is thoughtful, allowing students to learn and internalize the process deeply. Remember, each step builds upon the last, so clarity at every stage is crucial for accuracy.