Problem 53 - Algebra 1

Question

Evaluate the expression for the given value(s) of the variable(s). $$\frac{3 r-7}{11} \text { when } r=17$$

Step-by-Step Solution

Answer

The evaluated expression \(\frac{3 r-7}{11}\) for \(r=17\) is 4.

Step 1: Substitution

In the given expression \(\frac{3 r-7}{11}\), substitute the given value of \(r\) which is 17.

Step 2: Numerical Calculation

After substituting \(r = 17\), the expression becomes \(\frac{3*17-7}{11}\) which simplifies to \(\frac{51-7}{11}\), and further simplifies to \(\frac{44}{11}\).

Step 3: Simplify the Expression

The result of the simplification of \(\frac{44}{11}\) is 4.

Key Concepts

Substitution in Algebra

Substitution is a fundamental concept in algebra where you replace a variable with its given value. This method is like swapping out a placeholder for the actual number it represents. For example, if an algebraic expression contains a variable, say r, and you are told that r equals 17, you would replace every instance of r in the expression with the number 17.

This technique is essential for evaluating expressions because it transforms an abstract algebraic formula into a more concrete numerical form that can be calculated. Consider our textbook exercise where we are asked to evaluate \(\frac{3r - 7}{11}\) for r = 17. Substitution is the first step; by inserting 17 in place of r, the expression becomes calculable. It's a simple yet powerful tool that turns the unknown into a known, setting the stage for further mathematics to take place.

Simplifying Expressions

After you've substituted the variables with numbers, the next step in the process is to simplify the expression. Simplifying is all about breaking down the expression into its simplest form or the most reduced version of itself. This could involve combining like terms, factoring, expanding expressions, or, as in our example, performing the order of operations to carry out arithmetic calculations.

Simplification makes the expression easier to understand and often more practical to use, especially in applied mathematics or real-world scenarios. In our textbook problem, once we have replaced r with 17 and carried out the multiplication and subtraction, we simplify \(\frac{51-7}{11}\) to \(\frac{44}{11}\). Here, simplification means performing the division to find that the expression equals 4. The skill lies in recognizing that \(\frac{44}{11}\) is not just a fraction but represents a calculable number.

Numerical Calculations

Numerical calculations are the bedrock of working through algebra problems and evaluating expressions. These are the arithmetic operations you perform such as addition, subtraction, multiplication, and division once you've substituted variables and simplified the expressions as needed.

In our example, the calculation involves multiplying 3 by 17, subtracting 7 from the product, and then dividing the result by 11. This turns the algebraic expression into a numerical value of 4. By practicing numerical calculations, students enhance their computational fluency and their ability to work with numbers mentally or on paper, and in the case of more complex expressions, with calculators. These calculations are crucial because they give definitive answers to algebraic problems, answers that can be interpreted in the context of a given problem, such as in science, engineering, economics, or even day-to-day life.