Problem 25 - Algebra and Trigonometry

Question

Contain linear equations with constants in denominators. Solve each equation. $$\frac{x+3}{6}=\frac{3}{8}+\frac{x-5}{4}$$

Step-by-Step Solution

Answer

The solution to the equation is \(x=16.5\).

Step 1: Multiply by Common Denominator

Find the least common denominator (LCD) for all fractions. The LCD in this case is 24. We multiply both sides of the equation by 24 to eliminate the fractions: \(24*\frac{x+3}{6} = 24*\frac{3}{8} + 24*\frac{x-5}{4}\). Which simplifies to \(4(x+3)=3*3+6(x-5)\).

Step 2: Simplify the Equation

Expand and simplify the equation: \(4x+12=9+6x-30\). Which simplifies further to \(4x+12=6x-21\).

Step 3: Rearrange the Equation

Subtract 4x from both sides: \(12=2x-21\). Adding 21 to both sides gives: \(33=2x\).

Step 4: Solve for x

Finally, divide both sides of the equation by 2 to solve for x: \(x=\frac{33}{2}=16.5\).

Key Concepts

Least Common Denominator

When working with fractions in linear equations, we often encounter different denominators. To simplify the process of solving these equations, we need to find a common denominator, known as the Least Common Denominator (LCD). The LCD is the smallest number that each of the denominators can divide into without leaving a remainder. Finding the LCD helps us eliminate fractions and simplifies the equation, making it easier to solve.

• First, list all the denominators given in the equation. In this example, the denominators are 6, 8, and 4.
• Identify the smallest number that each of these denominators can be multiplied into. Here, that number is 24.

Using the LCD, we multiply each term in the equation by this number to clear the fractions, turning them into whole numbers and simplifying the solving process.

Fractions

Fractions represent parts of a whole and are commonly used in equations. In this exercise, we have:

• \( \frac{x+3}{6} \)
• \( \frac{3}{8} \)
• \( \frac{x-5}{4} \)

They signify division, where the numerator (the top number) is divided by the denominator (the bottom number). Fractions can complicate solving equations if they're not managed properly. This is why finding the least common denominator is crucial.
By multiplying each part of an equation by the LCD, we transform these fractions into whole numbers, simplifying the arithmetic involved. This step drastically reduces errors and simplifies the problem-solving process.

Solving Equations

Solving equations involves finding the value of the unknown variable that makes the equation true. In our example, the unknown variable is \( x \). The process begins by eliminating fractions through manipulation with the LCD, making calculations straightforward. For each transformed fraction:

• Multiply both sides of the equation by the LCD, turning the equation into one with integer coefficients.
• Simplify each term: \( 24*\frac{x+3}{6} = 4(x+3) \), and similarly for other terms.

Next, rearrange the equation to isolate the terms with the variable on one side and constant terms on the other. By simplifying and reorganizing the equation, we steadily approach the solution for \( x \). Finally, solve for \( x \) using arithmetic operations like addition, subtraction, multiplication, or division.

Simplification Steps

Simplifying an equation means to combine like terms and eliminate unnecessary complexities, making it easier to solve for the variable. After converting fractions to integers using the LCD, the next step is simplification:

• Expand and combine like terms: For example, expand \( 4(x+3) \) to \( 4x + 12 \).
• Simplify both sides of the equation step-by-step: \( 4x+12=6x-21 \).

Rearranging the equation helps further simplify it. Subtract \( 4x \) from each side to collect \( x \) terms together, resulting in a simpler equation: \( 12 = 2x - 21 \). Proceed with arithmetic operations such as adding onto both sides or dividing to isolate \( x \). Concluding with \( x = \frac{33}{2} \), these simplification steps ensure you arrive at an accurate and precise solution.