Problem 55 - Algebra 1

Question

You decide to see if you can ride the elevator to street level (Floor 0) without pushing any buttons. The elevator takes you up 4 floors, down 6 floors, up 1 floor, down 8 floors, down 3 floors, up 1 floor, and then down 6 floors to street level. Write and solve an equation to find your starting floor.

Step-by-Step Solution

Answer

The starting floor is the 7th floor.

Step 1: Represent the Elevator Movements

First, we represent each floor movement. An upward movement in the elevator is represented as a positive number, while a downward movement is represented as a negative number. The upward movements are: +4 floors and +1 floor (twice), and the downward movements are: -6 floors, -8 floors, -3 floors, and -6 floors.

Step 2: Set up the Equation

Next, we set up the equation. In mathematical terms, starting floor + total movements = 0 (street level). We know all the movements, the only unknown is the starting floor which we can represent as \(x\). So, the equation is: \(x + (4 -6 +1 -8 -3 +1 -6) = 0\)

Step 3: Solve the Equation

Now, solve the equation for \(x\), which is the starting floor. First, add all the numbers in the parenthesis to get the total movements: \(x + (-7) = 0\). Now solve for \(x\) by adding 7 to both sides of the equation: \(x = 0 + 7\).

Key Concepts

Integer Operations

Integer operations are essential in simplifying math problems involving directional movements. Here, integers represent elevator ascents and descents. Positive integers denote upward movements, while negative integers denote downward movements.
When faced with multiple successive integers, it's useful to add them together to obtain a single value representing the net movement.

• Add all positive values together to get the total upward movement.
• Add all negative values together or simply add the negative numbers to find the total downward movement.

For the elevator problem, integers are combined based on their direction:

• Up: +4, +1, +1
• Down: -6, -8, -3, -6

A clear understanding of integer addition allows us to solve problems with multiple directional steps effectively.

Solving Linear Equations

Linear equations are fundamental to algebra. A linear equation is an equation in which the highest power of the variable is one. In the elevator problem, we express the net result of movements as a linear equation.
The essential parts of a linear equation include:

• A variable (like \(x\), representing an unknown value).
• Constants (the total of movements).
• An equality sign (indicating that both sides are equal).

The given movements translate into the equation: \(x + (4 - 6 + 1 - 8 - 3 + 1 - 6) = 0\). Our goal is to solve for the variable \(x\), signifying the starting floor. Calculating the parenthesis provides the net movement, simplifying our equation to \(x - 7 = 0\). Isolating \(x\) results in \(x = 7\), indicating the elevator was initially on the 7th floor.

Problem Solving in Algebra

Problem solving in algebra requires identifying what is being asked, translating words into mathematical expressions, and then solving the problem step by step.
For the elevator question:

• Understand the scenario, where different movements are expressed as integers.
• Convert the real-world situation into a math equation representing the starting floor as \(x\).
• Consistently follow the operations order to simplify within the equation.

One effective strategy is to write down what you know and what you're trying to find. For example, you knew the elevator ended at street level and you wanted to find the starting floor.
Moreover, checking your solution helps confirm the answer. Add back movements to check if \(x = 7\) reaches floor 0. Good algebra problem solving enhances logical thinking, making future challenges easier to tackle.