Problem 11 - Algebra 1: Concepts and Skills

Question

Find the quotient. Evaluate \(\frac{a-4}{b}\) when \(a=-2\) and \(b=-3\)

Step-by-Step Solution

Answer

The evaluated quotient of the given expression is 2.

Step 1: Substitution

Replace 'a' with -2, and 'b' with -3 in the given expression to obtain \(\frac{-2-4}{-3}\)

Step 2: Perform Subtraction Operation

Carry out the subtraction operation in the numerator to get \(\frac{-6}{-3}\)

Step 3: Division Operation

Carry out the division operation to evaluate the expression. As the division of minus by minus is plus, the quotient between -6 and -3 equals to 2.

Key Concepts

Substitution

Substitution is a fundamental concept in algebra that helps us simplify or solve expressions by replacing variables with given values. It allows us to evaluate algebraic expressions in a specific context. When given an expression such as \( \frac{a-4}{b} \) and specific values for \( a \) and \( b \), you substitute these numbers into the expression.

For example, in the exercise, we substitute \( a = -2 \) and \( b = -3 \) into the expression, transforming \( \frac{a-4}{b} \) into \( \frac{-2-4}{-3} \).

• Identify the variables in the expression.
• Replace each variable with its given value.
• Ensure you are careful with sign changes when substituting negative numbers.
• Maintain the structure of the expression while you substitute the values.

Substitution is straightforward once you practice, making algebraic expressions easier to handle and evaluate!

Fraction Evaluation

Fraction evaluation involves performing arithmetic operations within a fractional setup, including subtraction in the numerator or denominator. In our expression \( \frac{-2-4}{-3} \), after substitution, our task is to perform any operations required to simplify this fraction.

Here, the subtraction in the numerator is calculated as follows:

• Combine like terms: Add or subtract the terms as needed.
• Handle negative numbers carefully: Remember the rules for subtracting negative values correctly.
• In the numerator \( -2-4 \), observe that subtracting 4 from -2 results in -6.

Proceeding in this manner with good attention to detail ensures a correct evaluation of fractions within algebraic expressions.

Division Operations

Division in algebra involves finding how many times one number is contained within another. This operation is not just about numbers but also about understanding how signs affect the result. In the final part of our expression \( \frac{-6}{-3} \), we need to divide \(-6\) by \(-3\).

Here are some key points to consider as you perform division in algebra:

• Determine the signs: Division with the same sign (negative and negative) results in a positive quotient.
• Calculate the magnitude: Ignore the sign initially, focus on dividing the absolute values.
• Conclude with the correct sign: Since both numbers are negative, the answer \(2\) is positive.

Algebraic division may occasionally seem tricky, but remember, handling signs correctly is crucial to arriving at the right answer!