Problem 20 - Algebra A Combined Function

Question

Solve each equation. See Examples 3 through \(5 .\) $$ \frac{2}{9} x-\frac{1}{3}=1 $$

Step-by-Step Solution

Answer

The solution is \( x = 6 \).

Step 1: Clear Fractions

Multiply every term in the equation by 9 to eliminate the fractions: \[ 9 \cdot \left( \frac{2}{9}x \right) - 9 \cdot \left( \frac{1}{3} \right) = 9 \cdot 1 \]This simplifies to:\[ 2x - 3 = 9 \]

Step 2: Add to Both Sides

Add 3 to both sides of the equation to isolate the term with the variable on one side:\[ 2x - 3 + 3 = 9 + 3 \]This simplifies to:\[ 2x = 12 \]

Step 3: Solve for x

Divide both sides by 2 to solve for \( x \): \[ \frac{2x}{2} = \frac{12}{2} \]This simplifies to:\[ x = 6 \]

Key Concepts

Clearing Fractions

When solving linear equations that include fractions, the first step is often to clear these fractions to make the equation simpler to manage. This is done by finding a common denominator or multiplying all terms in the equation by a number that will eliminate the fractions.
In the given example, we have the equation \( \frac{2}{9}x - \frac{1}{3} = 1 \). By multiplying each term by 9, the smallest common multiple of the denominators, we effectively eliminate the fractions. This makes the equation much easier to deal with, as it transforms into \( 2x - 3 = 9 \).

• Identifying the Common Multiple: Look at all the denominators in the equation.
• Multiplying Each Term: Multiply every term by the common multiple to clear fractions.
• Verification: Re-check your resulting equation to ensure no fractions remain!

Isolation of Variables

Once fractions are cleared, the next step is to isolate the variable. This means getting the variable you are solving for on one side of the equation by itself. For the example equation \( 2x - 3 = 9 \), this involves moving all terms that do not contain the variable to the other side of the equation.
To isolate \( x \), you would add 3 to both sides, removing the \( -3 \) from the left side. The equation then becomes \( 2x = 12 \). This process keeps the equation balanced, as whatever you do to one side must be done to the other.

• Add or Subtract: Use addition or subtraction to move constants from the variable's side.
• Balance the Equation: Maintain balance by applying changes to both sides.

Algebraic Manipulation

Once the variable is isolated to one side, algebraic manipulation is usually the last step needed to solve the equation. This involves operations such as division or multiplication to finally isolate the variable completely.
In the simplified equation \( 2x = 12 \), the coefficient of \( x \) is 2. To solve for \( x \), divide both sides by 2, giving you \( x = 6 \). This straightforward division isolates \( x \), solving the equation entirely.

• Divide or Multiply: Remove any coefficients using division or multiplication as needed.
• Double-Check: Always substitute your solution back into the original equation to verify accuracy.