Problem 86 - Beginning and Intermediate Algebra

Question

Simplify each expression. \(-\frac{7}{5}(t-15)-\frac{1}{2} t\)

Step-by-Step Solution

Answer

The simplified expression is \(-\frac{19}{10} t + 21\).

Step 1 - Distribute the fraction

Distribute \(-\frac{7}{5}\) to both terms inside the parentheses: \(-\frac{7}{5}(t-15) = -\frac{7}{5} t + \frac{7}{5} \times 15\).

Step 2 - Simplify the distribution

Calculate \(\frac{7}{5} \times 15 \): \(\frac{7}{5} \times 15 = \frac{7 \times 15}{5} = 21 \). Therefore, \(-\frac{7}{5}(t-15) = -\frac{7}{5} t + 21 \).

Step 3 - Rewrite the expression

Rewrite the original expression by substituting the distributed terms: \(-\frac{7}{5} t + 21 - \frac{1}{2} t \).

Step 4 - Combine like terms

Combine the like terms \(-\frac{7}{5} t \) and \(-\frac{1}{2} t \) by finding a common denominator: \[ -\frac{7}{5} t - \frac{1}{2} t = -\frac{14}{10} t - \frac{5}{10} t = -\frac{19}{10} t. \] So the simplified expression is \(-\frac{19}{10} t + 21\).

Key Concepts

combining like terms

In algebra, combining like terms is an essential skill. Like terms are terms that have the same variables raised to the same power. For instance, in the expression \( -\frac{7}{5} t - \frac{1}{2} t \), both terms contain the variable \( t \) and can be combined. This is possible because they only differ in their coefficients (the numerical part that multiplies the variables).
To combine like terms:

• Identify terms with the same variables and exponents.
• Add or subtract their coefficients.
• Keep the variable and exponent the same.

One way to easily combine like terms, as seen in the solution above, is by finding a common denominator if the terms have different denominators. Here, the terms \( -\frac{7}{5} t \) and \( -\frac{1}{2} t \) are combined by first converting them to have the same denominator. This combines them into one algebraic term: \( -\frac{14}{10} t - \frac{5}{10} t = -\frac{19}{10} t \). This process simplifies the overall expression and makes it easier to work with. [Combining Like Terms] can sometimes seem tricky, but with practice, it becomes easier to identify and simplify expressions efficiently.

• Always double-check that terms share both the same variables and exponents.
• Use common denominators where necessary to simplify fractions.

distributive property

The distributive property is a fundamental property for simplifying algebraic expressions. It states that a term multiplied by a sum (or difference) inside parentheses is equal to the sum (or difference) of the products of the term and each addend inside the parentheses.
For instance, in the expression \( -\frac{7}{5}(t - 15) \), we distribute \( -\frac{7}{5} \) to both \( t \) and \( -15 \). This means:
\[ -\frac{7}{5}(t - 15) = -\frac{7}{5} t + \frac{7}{5} \times 15 \]

• First, multiply the coefficient outside the parentheses with each term inside.
• Ensure to keep track of negative signs to avoid errors.

After distributing in the example above, we simplify by performing the multiplication: \[ -\frac{7}{5} \times 15 = 21 \]
This makes the expression: \[ -\frac{7}{5} t + 21 \]
The distributive property helps break down complex expressions into simpler parts, making them easier to simplify further.

• Always distribute to each term within the parentheses carefully.
• Simplify the resulting terms before combining like terms if possible.

common denominators

A common denominator is crucial when dealing with fractions, especially when combining like terms with fractional coefficients. To add or subtract fractions, they must have the same denominator. This unified denominator allows us to combine the numerators directly.
For example, to combine \( -\frac{7}{5}t \) and \( -\frac{1}{2}t \), we first need a common denominator.

• Identify the least common denominator (LCD) of the fractions involved. Here, the LCD of 5 and 2 is 10.
• Convert each fraction to an equivalent fraction with the LCD. \[ -\frac{7}{5} = -\frac{7 \times 2}{5 \times 2} = -\frac{14}{10} \] \[ -\frac{1}{2} = -\frac{1 \times 5}{2 \times 5} = -\frac{5}{10} \]
• Combine the fractions now that they share a common denominator: \[ -\frac{14}{10}t - \frac{5}{10}t = -\frac{19}{10}t \]

Finding a common denominator can be tricky at first, but it's a valuable technique and makes simplifying complex fractions straightforward.

• Practice finding the least common denominator with different fractions.
• Ensure the fractions are correctly converted to have the same denominator before combining them.