Problem 62 - Algebra 1: Concepts and Skills

Question

You decide to try to 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 floor 17.

Step 1: Identify and Write the Equation

The movements can be represented as follows: The elevator starts from an unknown floor \(x\), goes up 4 floors (\(+4\)), down 6 floors (\(-6\)), up 1 floor (\(+1\)), down 8 floors (\(-8\)), down 3 floors (\(-3\)), up 1 floor (\(+1\)), and then down 6 floors (\(-6\)) to reach the street level which is floor 0. Given these movements the equation representing this situation is: \(x + 4 - 6 + 1 - 8 - 3 + 1 - 6 = 0\).

Step 2: Simplify the Equation

Combine all the similar terms to simplify the equation: \(x - 6 + 1 - 3 - 6 = 0\) which simplifies further to \(x - 17 = 0\).

Step 3: Solve for the Unknown

To isolate \(x\), add 17 to both sides of the equation to get: \(x = 17\).

Key Concepts

Algebraic Expressions

Algebraic expressions are the foundational blocks of algebra. They help us represent numbers and operations in a symbolic form. In the scenario with the elevator, the expression is an equation that models the elevator's journey through the floors. An algebraic expression consists of numbers, variables, and the arithmetic operations of addition, subtraction, multiplication, or division. For example, in the equation from our problem, the variable \(x\) represents the unknown starting floor, while the coefficients and constants correspond to the floors the elevator rises or descends. Each movement of the elevator transforms the expression by adding or subtracting a constant value to/from \(x\). This abstraction allows for problem-solving through calculations and logical deductions, key tools in mathematics.

Linear Equations

Linear equations are equations of the first degree, which means they have variables raised to the power of one. They are called linear because their graph is a straight line when plotted on a coordinate plane.
For instance, the equation from the elevator problem, \(x - 17 = 0\), represents a linear relationship between \(x\) and the constant number 17. The goal in solving a linear equation is to determine the value of the unknown variable(s) that make the statement true. A linear equation typically takes the form \(ax + b = c\), where \(a\), \(b\), and \(c\) are constants, and \(x\) is the variable representing the unknown value. Solving linear equations involves isolating the variable on one side to determine its value.

Equation Solving Steps

The process of solving an equation involves several logical steps, similar to the methodical approach taken in the elevator exercise.
Here's what you need to do to solve such typical problems:

• **Identify the equation**: First, express the given problem in a mathematical equation using the correct operators and symbolize any unknown quantities with a variable, as shown in the exercise where the starting floor is represented by \(x\).
• **Simplify the equation**: Group and combine like terms to simplify the expression. This step reduces the complexity of the equation, as demonstrated by combining all numerical values and coefficients affecting \(x\). In our example, this simplification led to the equation \(x - 17 = 0\).
• **Solve for the variable**: Isolate the variable by performing operations that "undo" those in the equation, aiming to get the variable by itself on one side. In the step-by-step solution, we added 17 to both sides to find that \(x = 17\), giving us the answer.

These systematic steps ensure clarity and precision, allowing students to tackle any linear equation confidently.