Problem 15 - College Algebra with Modeling and Visualization

Question

Solve the equation and check your answer. $$ -5 x+3=23 $$

Step-by-Step Solution

Answer

The solution is \(x = -4\).

Step 1: Isolate the Variable

Start by subtracting 3 from both sides of the equation to help isolate the term with the variable on one side. This leads to the equation:\[-5x + 3 - 3 = 23 - 3\]which simplifies to:\[-5x = 20\]

Step 2: Solve for the Variable

Now divide both sides of the equation by -5 to solve for \(x\):\[x = \frac{20}{-5}\]Calculating, we find:\[x = -4\]

Step 3: Verify the Solution

To verify the solution, substitute \(x = -4\) back into the original equation and check:Substitute: \(-5(-4) + 3\)This simplifies to:\[20 + 3 = 23\]Since both sides of the equation match (23 = 23), the solution \(x = -4\) is correct.

Key Concepts

Isolating the Variable

In solving linear equations, isolating the variable is often the first critical step. The goal is to have the variable alone on one side of the equation, separated from other numbers. This involves performing operations that will systematically remove any numbers or operators that are with the variable. Let's take a closer look with the equation \(-5x + 3 = 23\):

• First, identify terms that are on the same side as the variable. Here, it's the \(+3\).
• To eliminate \(+3\), perform the inverse operation. This means you subtract \(3\) from both sides of the equation.
• By doing this, you ensure the equation remains balanced.

After subtracting \(3\), the equation simplifies to \(-5x = 20\). This correctly positions us to solve for \(x\) without any additional coefficients affecting the result. Keep in mind, isolating the variable is crucial for accurate calculation in any linear equation.

Equation Verification

Equation verification is the process of ensuring that the solution derived from a mathematical equation is accurate. It’s a simple yet essential final step because it confirms the correctness of your solution. Let’s explore this with our example:

After solving the equation \(-5x + 3 = 23\) and finding \(x = -4\), it's crucial to verify that this solution satisfies the original equation. Here’s how:

• Substitute the obtained value \(x = -4\) back into the original equation.
• Calculate the left side: \(-5(-4) + 3\).
• This simplifies to \(20 + 3 = 23\).

Since both sides of the equation match, it confirms the solution \(x = -4\) is correct. Verification is crucial, as it safeguards against any errors made in arithmetic or algebra. It is a good practice to always verify your solutions, ensuring accuracy.

Step-by-Step Solutions

A step-by-step approach is essential to solving equations comprehensively and accurately. This involves breaking down each action into smaller, manageable pieces, so let's revisit the example:

1. **Identify Steps Clearly** - Understand what each step is meant to achieve. For example, subtracting \(3\) from both sides was done to help isolate the variable.2. **Perform Accurate Calculations** - Each calculation should be checked as you go. This involves basic arithmetic in our equation \(-5x + 3 = 23\), like subtraction and division.3. **Use Logical Sequence** - Follow each operation in a logical order; flipping steps can lead to errors. In this equation, isolating was necessary before solving the value of \(x\).This structured breakdown not only makes solving linear equations easier but also builds a strong foundation for tackling more complex problems. In education, nurturing this methodical approach equips students with problem-solving confidence and skill.