Problem 10 - Algebra 2 and Trigonometry

Question

In \(3-20,\) solve each equation and check. $$ 0.2 a=0.05 a+3 $$

Step-by-Step Solution

Answer

The solution is \(a = 20\).

Step 1: Simplify the Equation

First, let's simplify the equation by isolating terms involving \(a\). Start by subtracting \(0.05a\) from both sides of the equation: \[ 0.2a - 0.05a = 3 \]This simplifies to: \[ 0.15a = 3 \]

Step 2: Solve for \(a\)

Next, solve for \(a\) by dividing both sides of the equation by \(0.15\): \[ a = \frac{3}{0.15} \]Perform the division: \[ a = 20 \]

Step 3: Check the Solution

To ensure the solution is correct, substitute \(a = 20\) back into the original equation and verify:\[ 0.2(20) = 0.05(20) + 3 \]Calculate both sides:\[ 4 = 1 + 3 \]Both sides are equal, confirming that \(a = 20\) is the correct solution.

Key Concepts

Equation Simplification

Equation simplification is an essential first step when solving linear equations. It involves manipulating the equation to make it as straightforward as possible, ideally with fewer and simpler terms. In the example given, you start with the equation:
\[ 0.2a = 0.05a + 3 \]
To simplify, you must focus on combining like terms and removing complexities, such as fractions or unnecessary symbols.

• Combine Like Terms: For instance, to simplify our equation, we start by subtracting \(0.05a\) from both sides. This operation helps isolate the variable, allowing us to gather similar terms, like \(a\), on one side:
  \[ 0.2a - 0.05a = 3 \]
  This results in a simpler equation:
• Clean Up: We've now reduced the equation to \(0.15a = 3\), making it much simpler. It is crucial in this step not to lose track of any terms or coefficients.

With the equation simplified, we're positioned perfectly to tackle the next core concept.

Variable Isolation

Variable isolation refers to the process of getting the variable alone on one side of the equation. This step reveals the value of the variable, which is our ultimate goal. Once you've simplified the equation as we did previously, the next challenge is to isolate \(a\).

• Identify the Operation: From \(0.15a = 3\), you see \(a\) is being multiplied by \(0.15\). To isolate \(a\), you need to perform the inverse operation to both sides of the equation.
  Divide: Simply divide both sides by \(0.15\):
  \[ a = \frac{3}{0.15} \]
• Complete the Calculation: When you perform this division, you'll find that \(a = 20\). Now, you have successfully isolated and solved for the variable \(a\).

Making the variable stand alone is crucial in ensuring you determine the correct answer. Every action performed on the equation must maintain the equality, paving the way to a verified solution.

Solution Verification

After isolating the variable and finding its value, verifying your solution ensures that it is indeed correct. Plugging the variable back into the original equation confirms this.

• Substitute Back: Use your found value in the original equation. For this exercise, substitute \(a = 20\) back into our starting point:
  \[ 0.2(20) = 0.05(20) + 3 \]
• Calculate Both Sides: By simplifying both sides of the equation, you confirm if both sides equal each other:
  \[ 4 = 1 + 3 \]
  When both calculations yield the same number, it verifies that our solution is accurate.

Verification is the final and crucial touch. It ensures that no mistakes were made during simplification or isolation. This step is like a proof of correctness, confirming the solution beyond any doubt.