Problem 4 - Algebra for College Students

Question

For Problems \(1-44\), solve each equation. $$ \frac{x+4}{3}-\frac{x-5}{9}=1 $$

Step-by-Step Solution

Answer

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

Step 1: Identify the Problem

The problem is to solve the equation \( \frac{x+4}{3} - \frac{x-5}{9} = 1 \). This requires finding the value of \( x \) that satisfies the equation.

Step 2: Clear the Fractions

To eliminate the fractions, we find a common denominator for the terms on the left-hand side. The denominators are 3 and 9, so the common denominator is 9. We'll multiply each term by 9 to clear the fractions: \[ 9 \left( \frac{x+4}{3} \right) - 9 \left( \frac{x-5}{9} \right) = 9 \times 1 \] which simplifies to \[ 3(x+4) - (x-5) = 9. \]

Step 3: Distribute and Simplify

Expand the expression: \[ 3(x+4) - (x-5) = 9 \] becomes \[ 3x + 12 - x + 5 = 9. \] Combine like terms: \[ (3x-x) + (12+5) = 9 \] which simplifies to \[ 2x + 17 = 9. \]

Step 4: Isolate the Variable

Subtract 17 from both sides to isolate terms with \( x \): \[ 2x + 17 - 17 = 9 - 17 \] which results in \[ 2x = -8. \]

Step 5: Solve for x

Divide both sides by 2 to solve for \( x \): \[ \frac{2x}{2} = \frac{-8}{2} \] So, \[ x = -4. \]

Step 6: Verify the Solution

Substitute \( x = -4 \) back into the original equation to check: \[ \frac{-4 + 4}{3} - \frac{-4 - 5}{9} = 1 \] Simplifies to \[ 0 - \frac{-9}{9} = 1 \] Which further simplifies to \[ 0 + 1 = 1. \] The solution is verified.

Key Concepts

Fractions in Equations

Fractions can make linear equations appear a bit more complex. Don't worry. Understanding them is quite straightforward. When you see an equation like \( \frac{x+4}{3} - \frac{x-5}{9} = 1 \), it essentially involves parts of quantities rather than whole quantities, divided by certain numbers. These numbers are called denominators. In the given equation, we have two fractions:

• \( \frac{x+4}{3} \)
• \( \frac{x-5}{9} \)

To solve equations with fractions, we look for ways to simplify or eliminate these fractions, often by finding a common multiple of the denominators. This makes the equation easier to handle.

Common Denominator

To manage equations that have fractions with different denominators, finding a common denominator is essential. This makes addition and subtraction straightforward. In the equation, our denominators are 3 and 9. The smallest number both divide into without leaving a remainder is 9. So, 9 is our common denominator. To clear the fractions, multiply each term by 9:

1. \( 9\left( \frac{x+4}{3} \right) \) results in \( 3(x+4) \)
2. \( 9\left( \frac{x-5}{9} \right) \) results in \((x-5)\)
3. \( 9 \times 1 \) becomes 9

This operation eliminates the fractions, resulting in an easier equation to solve.

Algebraic Manipulation

Once fractions are removed, we use algebraic manipulation to simplify the equation further and find the value of the variable. Begin by expanding and combining like terms: - Start with \( 3(x+4) - (x-5) = 9 \) and expand to \( 3x + 12 - x + 5 \).- Combine like terms: The terms \( 3x \) and \(-x\) simplify to \( 2x \). The constants 12 and 5 combine to give 17.This simplifies the equation to \( 2x + 17 = 9 \).Now, isolate \( x \) by subtracting 17 from both sides:- \( 2x = -8 \)Finally, solve for \( x \) by dividing both sides by 2:- \( x = -4 \) is the solution. These steps are fundamental in solving equations and form the crux of algebraic manipulation practices.

Verifying Solutions

Verifying your solution in algebra ensures your solution is correct. Substitute the value back into the original equation and simplify it to see if it holds true. For our example with \( x = -4 \), substitute back: \[ \frac{-4 + 4}{3} - \frac{-4 - 5}{9} = 1 \]Simplifies to:- \( 0 - \frac{-9}{9} = 1 \)or:- \( 0 + 1 = 1 \)Hence, the equation is satisfied. This step is crucial to confirm that no errors were made during the solving process. Always verify your solution to ensure it's correct. This not only builds confidence but also reinforces your understanding of the process.