Problem 89 - Beginning and Intermediate Algebra

Question

Simplify each expression. \(-\frac{2}{3}(5 x+7)-\frac{1}{3}(4 x+8)\)

Step-by-Step Solution

Answer

-\( \frac{14}{3} x - \frac{22}{3} \)

Step 1 - Distribute the Constants

Distribute \( -\frac{2}{3} \) to both terms inside the first parentheses: \( 5x \) and \( 7 \), and \( -\frac{1}{3} \) to both terms inside the second parentheses: \( 4x \) and \( 8 \). This results in: \[ -\frac{2}{3} \times 5x + (-\frac{2}{3} \times 7) - \frac{1}{3} \times 4x + (-\frac{1}{3} \times 8) \]

Step 2 - Perform the Multiplications

Calculate each multiplication: \[ -\frac{2}{3} \times 5x = -\frac{10}{3}x \] \[ -\frac{2}{3} \times 7 = -\frac{14}{3} \] \[ -\frac{1}{3} \times 4x = -\frac{4}{3}x \] \[ -\frac{1}{3} \times 8 = -\frac{8}{3} \]

Step 3 - Combine Like Terms

Combine the \( x \) terms: \[ -\frac{10}{3}x - \frac{4}{3}x = -\frac{14}{3}x \] Combine the constant terms: \[ -\frac{14}{3} - \frac{8}{3} = -\frac{22}{3} \] Therefore, the simplified expression is: \[ -\frac{14}{3}x - \frac{22}{3} \]

Key Concepts

distributive property

The distributive property helps us to multiply a number by a group of terms inside parentheses. It says that you can distribute the multiplication over addition or subtraction. In this exercise, we use the distributive property to simplify \( -\frac{2}{3}(5x+7)-\frac{1}{3}(4x+8)\).

We distribute \-\frac{2}{3}\ and \-\frac{1}{3}\ to each term inside the parentheses: \[ -\frac{2}{3} \times 5x + (-\frac{2}{3} \times 7) - \frac{1}{3} \times 4x + (-\frac{1}{3} \times 8) \]

This allows us to isolate the terms and make them simpler to work with. Remember that multiplying a fraction by a whole number means you multiply the numerator by the whole number and keep the denominator the same. That’s why \-\frac{2}{3} \times 5x\ turns into \-\frac{10}{3}x\ and so on.

combining like terms

After distributing the constants and performing the multiplications, we get several terms that are either multiples of \( x \) or constants. Combining like terms means we add or subtract the coefficients of terms that have the same variable.

In our exercise:

• First, combine the \( x \) terms: \[ -\frac{10}{3}x - \frac{4}{3}x = -\frac{14}{3}x \]
• Then, combine the constant terms: \[ -\frac{14}{3} - \frac{8}{3} = -\frac{22}{3} \]

Combining like terms makes the expression much simpler and easier to understand. Always look for terms with the same variable and combine them. And remember, keep an eye on the signs!

fractions in algebra

Fractions often make algebraic expressions look complicated, but they don’t have to be! Multiplying and adding fractions follows simple rules. When multiplying a fraction by a term, multiply the numerators and keep the denominator. For example: \[ -\frac{2}{3} \times 7 = -\frac{14}{3} \]

When adding or subtracting fractions, make sure the denominators are the same. In our exercise, the denominators were already the same, so we simply combined the numerators.

To summarize:

• Multiply the numerators when dealing with fractions and terms
• Keep the denominator the same
• Add or subtract the numerators directly when combining fractions with the same denominator

Understanding these rules can make fractions less intimidating and algebra much easier!