Problem 16 - College Algebra with Modeling and Visualization

Question

Solve the equation and check your answer. $$ -9 x-3=24 $$

Step-by-Step Solution

Answer

The solution is \(x = -3\).

Step 1: Move Constants to the Right Side

Start by moving the constant term on the left side of the equation to the right side. Add 3 to both sides to isolate the term with x on the left.\(-9x - 3 + 3 = 24 + 3\)This simplifies to:\(-9x = 27\)

Step 2: Isolate the Variable `x`

To solve for x, divide both sides of the equation by -9 to get x by itself:\(x = \frac{27}{-9}\)This simplifies to:\(x = -3\)

Step 3: Check Your Solution

Substitute \(x = -3\) back into the original equation to ensure it satisfies the equation:\(-9(-3) - 3 = 24\)\(27 - 3 = 24\)Since both sides equal 24, the solution is verified.

Key Concepts

Algebraic Manipulation

Algebraic manipulation is a crucial skill required to solve equations systematically. It involves rearranging, adding, subtracting, multiplying, or dividing parts of the equation to isolate the variable of interest. In algebra, our goal is often to solve for the unknown variable, such as \( x \). This process allows us to transform complex expressions into simpler ones.
To effectively perform algebraic manipulation, understanding the properties of equality is essential. Crucially, whatever operation you perform on one side of the equation must also be performed on the other side. This maintains the balance of the equation. For instance, to solve the equation \(-9x - 3 = 24\), we begin by adding 3 to both sides, hence maintaining equality and simplifying the equation step by step.
These operations help isolate variables and make equations easier and more intuitive to handle, which is a central goal in algebraic problem-solving.

Verification of Solution

Checking or verifying the solution is an essential step in solving equations, ensuring accuracy and reliability. Once you've found a solution, always substitute it back into the original equation. If it satisfies the equation, your solution is verified. This step checks against human error and verifies that no steps were missed or errors made during manipulation.
For example, after finding \(x = -3\) in the equation \(-9x - 3 = 24\), substitution back into the original equation was performed: \(-9(-3) - 3\). Simplifying this gives \(27 - 3\) which equals 24, matching the right side of the equation and confirming the solution is correct.
This verification provides confidence in your answer, ensuring consistency and correctness in mathematical calculations.

Linear Equation Steps

Solving linear equations involves sequential steps which, if followed, lead to the correct solution. The process is uniform, making it a predictable and reliable way to solve problems involving variables.
The steps begin with simplifying the equation as much as possible, often by eliminating constants and combining like terms. For the equation \(-9x - 3 = 24\), the first step involves moving the constant \(-3\) to the other side by adding 3 to both sides to maintain the balance.

• First, simplify: add or subtract terms from both sides to get the equation in a simpler form.
• Second, isolate the variable: divide or multiply to solve for the variable.
• Finally, verify: plug the variable back into the original equation to ensure no mistakes were made.

These steps are a crucial framework for solving not just specific problems, but a wide range of mathematical equations reliably and accurately.