Problem 15 - Algebra for College Students

Question

Solve each equation. \(\frac{x-2}{3}+\frac{x+3}{4}=\frac{11}{6}\)

Step-by-Step Solution

Answer

The solution to the equation is \( x = 3 \).

Step 1: Find a Common Denominator

To solve the equation, first find a common denominator for the fractions on both sides. The denominators are 3, 4, and 6. The least common multiple of these numbers is 12.

Step 2: Rewrite the Fractions

Multiply each term by the common denominator (12) to eliminate the fractions. \[ 12 \times \left( \frac{x-2}{3} \right) + 12 \times \left( \frac{x+3}{4} \right) = 12 \times \frac{11}{6} \] This simplifies to: \[ 4(x - 2) + 3(x + 3) = 22 \]

Step 3: Distribute and Simplify

Distribute the numbers across the parentheses: \[ 4x - 8 + 3x + 9 = 22 \] Combine like terms: \[ 7x + 1 = 22 \]

Step 4: Solve for x

Subtract 1 from both sides to isolate the term with x: \[ 7x = 21 \] Divide both sides by 7 to solve for x: \[ x = 3 \]

Key Concepts

Common Denominator

When dealing with equations involving fractions, finding a common denominator is crucial. In our equation, we have fractions with denominators 3, 4, and 6. The goal is to eliminate these fractions to simplify the process of solving the equation. To do this, we first find the least common multiple (LCM) of the denominators.

The denominators 3, 4, and 6 share a least common multiple of 12. This means we can multiply each term of the equation by 12, which simplifies our fractions.

• For the term \( \frac{x-2}{3} \), multiplying by 12 gives us \( 4(x-2) \).
• For \( \frac{x+3}{4} \), it becomes \( 3(x+3) \).
• Lastly, \( \frac{11}{6} \) turns into 22.

By transforming the equation with a common denominator, we simplify our work and make it easier to solve.

Distributive Property

The distributive property is a helpful tool in algebra that allows us to multiply a single term by two or more terms inside parentheses. After we've multiplied each side of our equation by 12 to eliminate fractions, we employ the distributive property.

In our simplified expression, we have:

• \( 4(x-2) \) which becomes \( 4x - 8 \).
• \( 3(x+3) \) translating to \( 3x + 9 \).

After applying the distributive property, our equation becomes \( 4x - 8 + 3x + 9 = 22 \). This step is crucial for removing the parentheses and getting closer to solving the equation.

Combining Like Terms

Once we apply the distributive property, our equation becomes a bit simpler, but we still need to tidy it up by combining like terms. Like terms are terms that have the same variable raised to the same power.

• In \( 4x - 8 + 3x + 9 \), we see terms involving \( x \): \( 4x \) and \( 3x \).
• We also have constant terms: -8 and +9.

By combining the like terms, we add the coefficients of \( x \) together: \( 4x + 3x = 7x \). We then combine the constants: \( -8 + 9 = 1 \). So, our equation simplifies to \( 7x + 1 = 22 \). Combining like terms reduces the equation to a much simpler form, making it easier to isolate the variable.

Isolating Variable

The final step in solving a linear equation is isolating the variable. In our equation \( 7x + 1 = 22 \), our goal is to solve for \( x \) by isolating it on one side of the equation.

To do this:

• First, subtract 1 from both sides of the equation: \( 7x + 1 - 1 = 22 - 1 \).
• This simplifies to \( 7x = 21 \).
• Finally, divide both sides by 7 to solve for \( x \): \( x = \frac{21}{7} \).
• This results in \( x = 3 \).

By subtracting and dividing, we successfully isolate \( x \). This process is important because it helps us find the specific value of \( x \) that solves the equation.