Problem 25 - College Algebra

Question

Exercises \(17-30\) contain 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 \(x=\frac{33}{2}\).

Step 1: Simplify the equation

Multiply each term in the equation by the least common denominator, which is 24. This gives, \[24*(\frac{x+3}{6})=24*(\frac{3}{8})+24*(\frac{x-5}{4})\]

Step 2: Solve the equation

Solving the equation gives, \(4(x+3)=9+6(x-5)\)

Step 3: Distribute and combine like terms

Distribute 4 and 6 to the terms within parentheses and you get, \(4x+12=9+6x-30\). Then, rearrange terms to one side the equation to get \(2x=33\).

Step 4: Solve for x

Finally, to isolate x, divide both sides of the equation by 2. This gives, \(x=\frac{33}{2}\).

Key Concepts

Solving Equations

Solving equations can seem daunting, but once you break them down step by step, it becomes more manageable. The goal is to find the value of the unknown variable, often expressed as 'x', that makes the equation true. In this particular exercise, we started with a rational equation which involves fractions. Here's a simple breakdown:

• First, simplify the equation as much as possible.
• Next, rearrange terms to get all terms containing the variable on one side of the equation and constants on the other.
• Simplify further if possible, to easily determine the solution.

This methodical process makes it easier to handle more complex equations, gradually working step-by-step to solve for the variable.

Least Common Denominator

When dealing with rational equations, combining fractions efficiently is key to finding solutions swiftly. To do this, we use the Least Common Denominator (LCD). The LCD is the smallest number that is a multiple of each of the denominators in the equation. It's like finding a common floor to bring all the fractions down to.

• Identify the denominators in your equation. For example, in our problem, we had 6, 8, and 4.
• Determine the smallest number that is evenly divisible by all these denominators. Here, the LCD is 24.
• Multiply each term in the equation by the LCD to eliminate the fractions, simplifying further manipulation of the equation.

This technique transforms the rational equation into one involving whole numbers, simplifying the solving process significantly.

Distributing Constants

Once fractions are eliminated, you often face expressions that need distribution, a process where you multiply a constant across terms within parentheses. Let's illustrate how this works using our exercise:

• We had to distribute 4 across \(x + 3\), yielding 4x + 12.
• Similarly, 6 had to be distributed across \(x - 5\), resulting in 6x - 30.

This step ensures each term inside the parentheses is multiplied by the constant outside, maintaining the balance of the equation. Distributing is crucial for simplifying and eventually solving equations, allowing each part of the expression to be unfolded and worked on clearly.

Combining Like Terms

In the process of solving equations, combining like terms is like organizing your notes. It helps in simplifying an equation, preparing it for the final steps of solving. Here's how we did it:

• We aligned similar terms on both sides of the equation: constants with constants, variables with variables. This reduces complexity.
• Our equation, 4x + 12 = 9 + 6x - 30, became easier to manage. By subtracting 6x from both sides and simplifying constants, we derived 2x = 33.

This streamlines the equation, making it easier to isolate the variable. With practice, identifying and combining like terms becomes second nature, turning a complex-looking problem into something more manageable.