Problem 39 - Elementary Algebra Within Reach

Question

In Exercises 39-42, write an algebraic equation. Do not solve the equation. After your instructor added 6 points to each student's test score, your score is 94 . What was your original score?

Step-by-Step Solution

Answer

The equation representing the student's original score is \(x + 6 = 94\).

Step 1: Identify the Variable

First, define a variable x to represent the original score before any points were added. This variable is what we are trying to find.

Step 2: Set up the Equation

In the scenario, it was given that the instructor added 6 points to the original score, and the new score was 94. Therefore, we can translate this situation into the equation: \(x + 6 = 94\).

Key Concepts

Variable Identification

To solve any algebraic equation, the first fundamental step is identifying the variable. In this scenario, we are trying to find out the original score of a student before the instructor added extra points. A variable acts as a placeholder for unknown values that we aim to discover. Here, the letter \(x\) is chosen to represent the original score. It's important to remember:

• The variable must clearly represent what you're solving for.
• In algebra, variables are usually denoted by letters like \(x\), \(y\), or \(z\). But you can use any symbol that makes the equation easier to understand for you.
• Ensure everyone who sees your variable knows what it stands for in the context of the problem. In this example, \(x\) stands for "the original score."

Identifying the variable is like setting a goal in problem solving; it provides a clear direction for which unknown value you're trying to uncover.

Equation Setup

Once you've identified your variable, it's time to set up the equation. This involves expressing the scenario given in the problem using mathematical symbols and equations. In the exercise, we're told that after adding 6 points to the student's original score, the final score is 94. We can translate this into a mathematical equation by writing:\[ x + 6 = 94 \]Let's break this down:

• \(x\) represents the original score, which is what we're trying to find.
• The number 6 is added to \(x\), reflecting the 6 extra points that the instructor added.
• The result, 94, is the new score after the addition.

Setting up an equation correctly is crucial because it transforms the problem's words into a mathematical "story" that can guide us to the solution. The equation \(x + 6 = 94\) sums up all the information provided in a concise, mathematical format.

Mathematical Translation

Mathematical translation involves converting everyday language into mathematical equations, which is a key skill in solving algebra problems. It allows us to see logical relationships clearly and solve for unknowns using algebraic methods.For our problem:- We take the statement "after your instructor added 6 points, your score is 94" and turn it into an equation.- The phrase "added 6 points" becomes "\(+ 6\)."- "Your score is 94" is translated to "\( = 94\)."Understanding this process involves:

• Clearly interpreting language and numbers.
• Accurately using mathematical symbols.
• Constructively modeling real-world problems in a math context.

Mathematical translation not only frames the problem but also prepares it for algebraic manipulation, enabling us to reach our desired solutions efficiently. In this context, learning to read the "language of math" is like learning to interpret a map that guides us through the problem.