Problem 30 - Elementary and Intermediate Algebra

Question

Solve each equation. $$ \frac{2}{3}(x-3)=\frac{1}{6}(7 x+29)+3 $$

Step-by-Step Solution

Answer

The solution for the equation is \( x = -\frac{59}{3} \).

Step 1: Distribute and Simplify

Start by distributing the fractions on both sides of the equation: \( \frac{2}{3}(x-3) = \frac{2}{3}x - 2 \) and \( \frac{1}{6}(7x+29) = \frac{7}{6}x + \frac{29}{6} \). The equation becomes:\[ \frac{2}{3}x - 2 = \frac{7}{6}x + \frac{29}{6} + 3 \].

Step 2: Combine like terms on Right Side

Combine the constant terms on the right side: \( \frac{29}{6} + 3 \).Convert 3 to a fraction: \( 3 = \frac{18}{6} \).Add: \( \frac{29}{6} + \frac{18}{6} = \frac{47}{6} \).Substitute back: \( \frac{2}{3}x - 2 = \frac{7}{6}x + \frac{47}{6} \).

Step 3: Get Rid of Fractions

To eliminate the fractions, multiply every term by 6:\[ 6 \left( \frac{2}{3}x - 2 \right) = 6 \left( \frac{7}{6}x + \frac{47}{6} \right) \]This simplifies to:\[ 4x - 12 = 7x + 47 \].

Step 4: Solve for x

Subtract \( 7x \) from both sides: \[ 4x - 7x = 47 + 12 \]This simplifies to:\[ -3x = 59 \].Next, divide both sides by -3:\[ x = -\frac{59}{3} \].

Key Concepts

Fractions in Equations

When solving equations, fractions often make them appear more complicated than they really are. However, understanding and manipulating these fractions is essential. Let's break it down further.

• Fractions contain both a numerator and a denominator. In the equation given, \( \frac{2}{3}(x-3) \) and \( \frac{1}{6}(7x+29) \) already include variables with fractions.
• The goal is to simplify or isolate variables step by step, but dealing with fractions first often simplifies the entire equation.

Think of fractions as just another type of coefficient. By mastering how to interpret and distribute fractions accurately, you will find that linear equations become much less intimidating. Fractions can be managed with basic operations and by converting them to a common denominator when necessary.

Distributive Property

The distributive property is a powerful tool in algebra. It allows us to remove parentheses by distributing multiplication over addition or subtraction inside. In this exercise, we have:

• For the term \( \frac{2}{3}(x-3) \), the distribution means multiplying \( \frac{2}{3} \) by both \( x \) and \( -3 \), giving us \( \frac{2}{3}x - 2 \).
• Similarly, for \( \frac{1}{6}(7x+29) \), distribute \( \frac{1}{6} \) across \( 7x \) and \( 29 \) resulting in \( \frac{7}{6}x + \frac{29}{6} \).

By applying the distributive property, you can rewrite expressions so that the fractions and constants are clearly separated. This step is crucial before proceeding to solve the equation further.

Combining Like Terms

Once we have applied the distributive property, we often find ourselves with terms that need combining. Particularly, constant terms or terms solely consisting of numbers should be added together. In this exercise, consolidating the constants on the right side of the equation was necessary.

• Combine constants with common denominators: \( \frac{29}{6} + 3 \) where \( 3 \) is converted to \( \frac{18}{6} \) for easy addition.
• Sum them up to simplify: \( \frac{47}{6} \).

Combining like terms helps in simplifying the equation further, ensuring it is in its most elemental form before tackling solutions.

Eliminating Fractions

Fractions can make calculations cumbersome, so eliminating them when solving equations is advantageous. This process involves finding a common factor to multiply through, thereby clearing the fractions. Here’s how to deal with them:

• Multiply every term by the least common multiple of the denominators—in this case, 6. This simplifies the equation by transforming it into one without fractions.
• The equation then becomes: \( 4x - 12 = 7x + 47 \).

Clearing fractions simplifies the arithmetic as you solve for \( x \). Once fractions are removed, you proceed to standard algebraic manipulation to isolate the variable. Remember, finding the least common multiple is key to successfully eliminating fractions from linear equations.