Problem 83 - Intermediate Algebra

Question

Simplify. See Example \(6 .\) $$36\left(\frac{2}{9} x-\frac{3}{4}\right)+36\left(\frac{1}{2}\right)$$

Step-by-Step Solution

Answer

The simplified expression is \( 8x - 9 \).

Step 1: Distribute 36

The expression to simplify is \( 36\left(\frac{2}{9} x-\frac{3}{4}\right)+36\left(\frac{1}{2}\right) \). First, distribute \( 36 \) into both terms inside the parentheses.

Step 2: Multiply and Simplify Each Terms

Calculate \( 36 \times \frac{2}{9} x \) and \( 36 \times \left(-\frac{3}{4}\right) \). Starting with the first term: \( 36 \times \frac{2}{9} x = 8x \) and for the second term: \( 36 \times \left(-\frac{3}{4}\right) = -27 \).

Step 3: Calculate Additional Multiplication

Now, calculate \( 36 \times \frac{1}{2} \) which equals \( 18 \).

Step 4: Combine All Terms

Combine all terms from the multiplication: \( 8x - 27 + 18 \). Now, combine the constant terms: \( -27 + 18 = -9 \).

Step 5: Write Final Expression

The simplified expression is \( 8x - 9 \).

Key Concepts

Distributive Property

The distributive property is a fundamental algebraic concept that allows us to multiply a single term by each term in a sum or difference within parentheses. In essence, it "distributes" the multiplication across the terms inside the parentheses.

For example, when we have an expression like \( a(b + c) \), the distributive property tells us to multiply \( a \) by both \( b \) and \( c \). So, it becomes \( ab + ac \).

In the exercise \( 36\left(\frac{2}{9} x-\frac{3}{4}\right)+36\left(\frac{1}{2}\right) \):

• First, \( 36 \) is distributed to each term inside \( \left(\frac{2}{9} x-\frac{3}{4}\right) \) giving us \( 36\times\frac{2}{9} x \) and \( 36\times\left(-\frac{3}{4}\right) \).
• Then, \( 36 \) is also distributed to \( \left(\frac{1}{2}\right) \), simplifying that part separately.

Employing the distributive property correctly helps us to simplify complex expressions, step-by-step.

Combining Like Terms

Combining like terms is an essential process in simplifying algebraic expressions. This involves adding together terms that have the same variable raised to the same power.

Consider any expression: If you notice terms with the same variables, you can combine them by adding or subtracting their coefficients. For instance, \( 3x + 2x = 5x \) because they both have the same variable: \( x \).

In our specific example from the exercise, after distributing and multiplying, we are left with: \( 8x - 27 + 18 \). Here, the terms \(-27\) and \(+18\) are constants (they have no variable), so they can be combined:

• Combine \(-27\) and \(+18\) to get \(-9\).

Thus, the expression becomes \( 8x - 9 \) after combining like terms. Simplifying expressions by combining like terms makes them more concise and manageable.

Multiplication of Fractions

Understanding the multiplication of fractions is key when working with expressions that include fractional coefficients. The process involves multiplying the numerators and denominators separately.

Suppose we have a fraction \( \frac{a}{b} \) multiplied by another number. You multiply \( a \) by that number and keep \( b \) in the denominator, simplifying if possible.

In the exercise:

• When multiplying \( 36 \) by \( \frac{2}{9} \), you actually calculate \( \frac{36 \times 2}{9} \). Simplify to get \( 8 \), so the first term becomes \( 8x \).
• Similarly, for \( 36 \times \left( -\frac{3}{4} \right) \), it translates to \( \frac{36 \times (-3)}{4} \), simplifying to \(-27\).

This shows that being comfortable with fractions simplifies the process. Using multiplication and simplification allows us to work seamlessly with fractions in algebraic contexts.