Problem 40 - Algebra 1: Concepts and Skills

Question

Evaluate the expression for the given value(s) of the variable(s). \(\frac{3 r-3}{11}\) when \(r=-10\)

Step-by-Step Solution

Answer

Evaluating the given expression \(\frac{3 r-3}{11}\) for \(r = -10\) gives the result as -3

Step 1: Substitute the Given Value

Substitute \(r = -10\) into the expression \(\frac{3 r-3}{11}\). The resultant expression will be \(\frac{3 (-10) -3}{11}\).

Step 2: Solve the Equation

Perform the operations inside the parentheses first. Multiply three by negative ten to get negative thirty, subtract three from negative thirty to get \(-30 - 3 = -33\). Thus, you obtain \(\frac{-33}{11}\).

Step 3: Simplify the Equation

Now, divide -33 by 11 to get the result \(-33 / 11 = -3\).

Key Concepts

Substitute Values

Substituting values is the first and vital step in evaluating algebraic expressions. Essentially, it involves replacing a variable in an expression with a given number. In this exercise, we are given an expression with a variable \( r \), and we need to evaluate it for \( r = -10 \).

• Identify the variable in the given expression. In this case, it's \( r \).
• Replace \( r \) with \(-10\) within the expression \( \frac{3r - 3}{11} \).
• Once substituted, the expression becomes \( \frac{3(-10) - 3}{11} \).

Substitution is straightforward because it turns the expression into one that only contains numbers, making it easier to solve. The main goal of this step is to ensure that the variable disappears, simplifying the expression for further operations.

Simplify Expressions

After substituting a value into an algebraic expression, the next step is often to simplify it. Simplifying means making an expression easier to work with or understand while keeping its value the same. For the expression \( \frac{3(-10) - 3}{11} \) that arises after substitution:

• Conduct operations in the numerator: Begin by performing multiplication and subtraction in sequence.
• Multiply \( 3 \times (-10) \) to get \(-30\).
• Subtract 3 from \(-30\) to reach \(-33\).

The goal during simplification is to reduce the expression step by step until you end up with a single numeric fraction, ready for final division. It's akin to breaking down a task into smaller, manageable parts.

Arithmetic Operations

Once the expression is simplified, arithmetic operations come into play to calculate the final value. This process turns the problem into a simple divisional task. In our exercise, the expression is reduced to \( \frac{-33}{11} \):

• Division is the prominent operation here. It involves dividing the numerator by the denominator.
• Divide \(-33\) by \(11\). As you find \(-33 \div 11 = -3\).

Arithmetic operations include addition, subtraction, multiplication, and, as in this case, division. Completing this step correctly ensures you find the correct numerical result of the given expression.