Problem 90 - Beginning and Intermediate Algebra

Question

Simplify each expression. \(-\frac{3}{4}(7 x+9)-\frac{1}{4}(5 x+7)\)

Step-by-Step Solution

Answer

The simplified expression is \(-\frac{13}{2}x - \frac{17}{2}\).

Step 1: Expand the expressions

Distribute the fractions to each term inside the parentheses. First term: \(-\frac{3}{4}(7 x + 9) = -\frac{3}{4} \times 7 x - \frac{3}{4} \times 9 = -\frac{21}{4}x - \frac{27}{4}\)Second term: \(-\frac{1}{4}(5 x + 7) = -\frac{1}{4} \times 5 x - \frac{1}{4} \times 7 = -\frac{5}{4}x - \frac{7}{4}\)

Step 2: Combine like terms

Add the coefficients of like terms (terms with \(x\) together and constant terms together). \(-\frac{21}{4}x - \frac{5}{4}x = -\frac{26}{4}x = -\frac{13}{2}x \) \(-\frac{27}{4} - \frac{7}{4} = -\frac{34}{4} = -\frac{17}{2} \)

Step 3: Write the final simplified expression

Combine the results from the previous step to form the simplified expression.\(-\frac{13}{2}x - \frac{17}{2}\)

Key Concepts

distributive property

The distributive property is a useful tool in algebra that helps us simplify expressions involving parentheses. It states that for any numbers or expressions \(a\), \(b\), and \(c\), the expression \(a(b + c)\) can be expanded to \(ab + ac\). This means you multiply each term inside the parentheses by the term outside the parentheses.

Let's apply this to our exercise:
Starting with the expression \(-\frac{3}{4}(7 x+9)-\frac{1}{4}(5 x+7)\), we’ll use the distributive property to remove the parentheses. For the first term, distribute \-\frac{3}{4}\ to both \(7x\) and \(9\):
\[-\frac{3}{4}(7 x + 9) = -\frac{3}{4} \times 7 x - \frac{3}{4} \times 9 = -\frac{21}{4}x - \frac{27}{4}\]

Similarly, distribute \-\frac{1}{4}\ to both \(5x\) and \(7\):
\[-\frac{1}{4}(5 x + 7) = -\frac{1}{4} \times 5 x - \frac{1}{4} \times 7 = -\frac{5}{4}x - \frac{7}{4}\]

After distribution, the expression now looks like:
\[-\frac{21}{4}x - \frac{27}{4} - \frac{5}{4}x - \frac{7}{4}\]
This expansion simplifies our calculations in the next steps.

combining like terms

Once the distributive property has been applied, the next step is to combine like terms. Like terms are terms that have the same variable raised to the same power. For example, in the expression \(5x + 3x\), both terms are like terms, and can therefore be combined.

In our exercise, we have the expanded expression:
\[-\frac{21}{4}x - \frac{27}{4} - \frac{5}{4}x - \frac{7}{4}\]

Like terms here are the terms with \(x\) and the constant terms:

• \(-\frac{21}{4}x\) and \(-\frac{5}{4}x\) are like terms.
• \(-\frac{27}{4}\) and \(-\frac{7}{4}\) are also like terms.

Let's combine them:
Combine the coefficients of \(x\):
\[-\frac{21}{4}x - \frac{5}{4}x = -\frac{26}{4}x = -\frac{13}{2}x\]
Combine the constants:
\[-\frac{27}{4} - \frac{7}{4} = -\frac{34}{4} = -\frac{17}{2}\]

This step simplifies our expression to:
\[-\frac{13}{2}x - \frac{17}{2}\]

fraction operations

Finally, it’s important to understand fraction operations, which are crucial for simplifying algebraic expressions. Fractions often appear in algebra, and knowing how to handle operations like multiplication, addition, and subtraction with fractions is essential.

In our example, multiplication of fractions was used during the distributive property step.
To multiply fractions, multiply the numerators together and the denominators together:

• For instance, \( -\frac{3}{4} \times 7 = -\frac{3 \times 7}{4} = -\frac{21}{4} \)

When adding or subtracting fractions, the denominators must be the same. Once they are common, simply add or subtract the numerators:

• In our expression, \(-\frac{27}{4} - \frac{7}{4}\) could be combined because they already have a common denominator, resulting in \(-\frac{34}{4}\).
• This is simplified to: \( -\frac{34}{4} = -\frac{17}{2} \)

Mastering these fraction operations will make algebra much easier, especially when simplifying complex expressions involving multiple steps.