Problem 61 - Algebra A Combined Function

Question

Evaluate each expression when \(x=-5, y=4,\) and \(t=10\). \(\frac{9-x}{y+6}\)

Step-by-Step Solution

Answer

\( \frac{7}{5} \)

Step 1: Substitute the Given Values

Replace the variables in the expression \( \frac{9-x}{y+6} \) with the given values. This will give us: \( \frac{9-(-5)}{4+6} \).

Step 2: Simplify Inside the Numerator

Simplify the expression inside the numerator. \( 9-(-5) \) equals \( 9+5 \) because subtracting a negative is the same as adding. This becomes \( 14 \).

Step 3: Simplify Inside the Denominator

Simplify the expression inside the denominator. \( 4+6 \) equals \( 10 \).

Step 4: Write the Final Expression

Now plug the simplified values back into the expression: \( \frac{14}{10} \).

Step 5: Simplify the Fraction

Reduce the fraction \( \frac{14}{10} \) to its simplest form by dividing both the numerator and denominator by their greatest common divisor 2. This gives us \( \frac{7}{5} \).

Key Concepts

Substitution

Substitution is the core step in evaluating expressions. It involves replacing variables with given numerical values. In our example, we've been provided specific values for the variables in the expression. For the expression \( \frac{9-x}{y+6} \), we substitute \( x = -5 \), \( y = 4 \). This makes the expression \( \frac{9-(-5)}{4+6} \).

Substitution helps simplify the problem by converting algebraic expressions into purely numerical ones so you can compute them directly. This direct form allows us to work easily with basic arithmetic operations rather than juggling abstract variables. Remember, always pay clear attention to the signs of the numbers being substituted, as it can greatly affect the outcome.

Simplifying Fractions

Simplifying fractions is a process of making a fraction as simple as possible. In other words, this means writing it in its lowest terms by eliminating any common factors between the numerator and the denominator. This is achieved by dividing both the top and bottom by their greatest common divisor.

In our scenario, the fraction we end up with after substitution and computation is \( \frac{14}{10} \). By examining both the numerator and the denominator, we identify 2 as a common factor. Dividing both by 2 results in the simplified fraction \( \frac{7}{5} \).

• Always check whether any common factors exist between numerator and denominator.
• Simplifying fractions helps in keeping numbers manageable and easier to work with.

Numerator and Denominator

A fraction contains two parts: the numerator and the denominator. The numerator is the number above the fraction line, and it tells us how many parts we have. The denominator is below the line, indicating into how many equal parts the whole is divided.

For instance, in the fraction \( \frac{9-x}{y+6} \), the numerator is \(9-x\), and the denominator is \(y+6\). Together, these make up the entire value of the fraction. When simplifying, ensure that both are reduced to form the simplest possible fraction, maintaining the mathematical equivalence by sharing the same divisor.

Understanding the roles of these two components helps you manipulate and reduce fractions accurately during simplification.

Greatest Common Divisor

The greatest common divisor (GCD) of two numbers is the largest number that divides both of them without leaving a remainder. It's a vital concept when simplifying fractions as it allows us to reduce them to their smallest form.

For simplifying our fraction \( \frac{14}{10} \), we find that 2 is the GCD. Dividing both the numerator and denominator by 2 results in the simplest form, \( \frac{7}{5} \).

• To find the GCD, list the factors of both numbers and identify the largest common one.
• The GCD provides a way to ensure fractions are simplified completely and correctly.

Being comfortable with finding the GCD ensures you can simplify any fraction accurately, giving clarity and precision to your mathematical results.