Problem 14 - Elementary Algebra

Question

Solve each of the equations. $$\frac{x-2}{5}-\frac{x+3}{6}=-4$$

Step-by-Step Solution

Answer

The solution is \(x = -93\).

Step 1: Eliminate Fractions

To eliminate the fractions, find the least common denominator (LCD) for the denominators 5 and 6. The LCD is 30. Multiply each term by 30: \[30 \times \left( \frac{x-2}{5} \right) - 30 \times \left( \frac{x+3}{6} \right) = 30 \times (-4).\] This simplifies to: \[6(x-2) - 5(x+3) = -120.\]

Step 2: Distribute and Simplify

Distribute the numbers in front of the parentheses: \[6(x) - 6(2) - 5(x) - 5(3) = -120.\] This simplifies to: \[6x - 12 - 5x - 15 = -120.\] Combine like terms: \[x - 27 = -120.\]

Step 3: Solve for x

Add 27 to both sides to isolate \(x\): \[x - 27 + 27 = -120 + 27.\] This simplifies to: \[x = -93.\]

Key Concepts

Fractions in Algebra

Fractions are an important part of algebra. When we see fractions in an equation, they might look tricky, but with practice, they become easier to handle. Fractions indicate division of the numerator by the denominator, and managing them well is crucial in solving algebraic equations.
In algebra, fractions can represent parts of equations and expressions. Often, we need to manipulate these fractions to simplify the equation. For instance, the exercise above involves fractions like \( \frac{x-2}{5} \). The presence of fractions tells us that we need special strategies, like finding a common denominator or multiplying through to eliminate them. This will let us solve the equation more easily.

Least Common Denominator

The concept of the Least Common Denominator (LCD) is essential when dealing with fractions. The LCD is the smallest number that is divisible by each of the denominators in the fractions of an equation. In our exercise, the denominators are 5 and 6.
To eliminate fractions from the equation, we find the LCD of 5 and 6, which is 30. By multiplying every term by the LCD (30 in this case), the fraction parts are cleared, making the equation simpler. Doing so transforms our equation

• from \( \frac{x-2}{5} - \frac{x+3}{6} = -4 \)
• to \( 6(x-2) - 5(x+3) = -120 \).

This step was crucial to getting rid of fractions and allowing us to work with whole numbers.

Distribution in Algebra

Distribution in algebra involves spreading out multiplication over terms inside parentheses. It's a way to simplify and solve equations by applying a multiply-then-add strategy.
For our equation \( 6(x-2) - 5(x+3) = -120 \), we distribute the 6 across \( (x-2) \) and the \(-5\) across \( (x+3) \):

• This gives us: \( 6 \times x - 6 \times 2 \) and \( -5 \times x - 5 \times 3 \).

These steps resolve into the individual terms \( 6x - 12 - 5x - 15 \), from which we can go on to simplify further by combining like terms.
The distributive property is fundamental, making complex equations easier to handle by breaking them down into manageable parts.

Combining Like Terms

After distribution, we often end up with terms that are similar, or like terms, which we can combine to simplify the equation further. Like terms are terms with the exact same variable part, such as \(6x\) and \(-5x\) in our equation.
By adding or subtracting these terms, we condense the equation:

• \( 6x - 5x \) simplifies to \( x \).
• The constants \(-12\) and \(-15\) combine to \(-27\).

So, our equation is simplified to \( x - 27 = -120 \). Combining like terms helps to neatly pack the equation, getting it ready for the final steps toward finding the solution. Applying this concept reduces complexity and makes solving for the variable, like \(x\), straightforward.