Problem 4 - Algebra A Combined Function

Question

Solve. For Exercises 1 through \(4,\) write each of the following as equations. The sum of 4 times a number and -2 is equal to the sum of 5 times the number and \(-2 .\) Find the number.

Step-by-Step Solution

Answer

The number is 0.

Step 1: Define the unknown variable

Let's define the unknown number as \( x \). This will help us create a mathematical equation to represent the problem.

Step 2: Translate words into a mathematical equation

According to the problem, the sum of 4 times the number and -2 is equal to the sum of 5 times the number and -2. We can translate this into the equation: \( 4x - 2 = 5x - 2 \).

Step 3: Simplify the equation

We observe that both sides of the equation have a \(-2\). Since subtracting 2 on both sides doesn't affect the equality, we can simplify the equation to: \( 4x = 5x \).

Step 4: Solve for the variable

To isolate the variable \(x\), subtract \(4x\) from both sides of the equation: \(4x - 4x = 5x - 4x\) which simplifies to \(0 = x\). Thus, the number is \(x = 0\).

Step 5: Verify the solution

Substitute \(x = 0\) back into the original expression. The left side \(4(0) - 2 = -2\) and the right side \(5(0) - 2 = -2\) are equal, verifying that the solution is correct.

Key Concepts

Equation Simplification

When solving linear equations, equation simplification is crucial. It helps in reducing complex expressions to simpler forms, making mathematical problems easier to tackle. In our example, we identified an equation: \(4x - 2 = 5x - 2\). To simplify, notice that both sides of the equation share a common term, \(-2\).

By removing such identical terms from both sides, it leads to a simplified, more manageable equation. Simplification is an essential step because:

• It minimizes mistakes by reducing unnecessary complexity.
• It helps focus on the variable of interest.
• It can reveal simpler relationships between terms.

In this exercise, once \(-2\) was eliminated from both sides, it left us with \(4x = 5x\). This shows the power of simplification in solving linear equations effectively.

Variable Isolation

Variable isolation is at the heart of solving equations. This means you alter the equation so that the unknown variable is on one side, making it easier to find its value. Once we have our simplified equation \(4x = 5x\), the next step is to isolate the variable \(x\).

To achieve this, subtract \(4x\) from both sides. By performing the same operation on each side, the equation remains balanced and we obtain \(0 = x\). This shows that our unknown variable is equal to zero. Key aspects of variable isolation include:

• Applying operations (addition, subtraction, multiplication, division) evenly on both sides.
• Converting complex expressions to simpler forms with less clutter.
• Ensuring the variable stands alone, facilitating straightforward solution.

Variable isolation enables you to identify the specific value a variable holds, which in turn provides concrete solutions to problems.

Verification of Solutions

Verification of solutions serves as a check post in problem-solving. It ensures that the solution we arrived at is indeed correct and satisfies the original equation. In our example, after isolating the variable, we found \(x = 0\). Verification involves substituting this back into the original equation.

Here, substituting \(x = 0\) results in:

• Left side: \(4(0) - 2 = -2\)
• Right side: \(5(0) - 2 = -2\)

Both sides equate to \(-2\), proving our solution correct. This process is crucial as it:

• Confirms the accuracy of calculations and simplifications.
• Validates the method used was correct.
• Provides confidence in the derived answer.

Verification ensures that each step taken in the problem-solving journey adds up, protecting against errors and yielding reliable solutions.