Problem 2 - Elementary Algebra

Question

Write "identity," "contradiction," or "conditional." If you can, find the solution by making an educated guess based on your knowledge of arithmetic. $$ y-4=7 $$

Step-by-Step Solution

Answer

Answer: The solution to the equation is $$y = 11$$.

Step 1: Identify the equation type

The given equation is a simple one-variable linear equation which can be written as: $$y - 4 = 7$$

Step 2: Add 4 to both sides of the equation

To isolate the variable, y, on the left side of the equation, we need to add 4 to both sides. This will eliminate the '-4' on the left side: $$y - 4 + 4 = 7 + 4$$

Step 3: Simplify both sides of the equation

After adding 4 to both sides, our equation becomes: $$y = 11$$ Now the variable is isolated and the value of y is found. So the solution to the equation $$y - 4 = 7$$ is $$y = 11$$.

Key Concepts

Linear Equations

Linear equations are mathematical statements involving equalities. They can be expressed in the form \( ax + b = c \), where \( x \) is the variable, and \( a \), \( b \), and \( c \) are constants.
A linear equation is "linear" because the variable is only raised to the power of one.
This means that graphically, if you were to plot these on an x-y coordinate plane, they would always be straight lines.

• Linear component: The variable (usually \( x \) or \( y \)) is only to the first power.
• Graph: Represents a straight line which shows a consistent rate of change.

In essence, linear equations represent relationships where one quantity depends on another, without involving powers or square roots.

Algebraic Operations

Algebraic operations are the basic computations used to manipulate equations. They encompass addition, subtraction, multiplication, and division among others.
These operations are crucial in solving equations as they help to isolate variables.
When we solve an equation, we use these algebraic operations to "undo" the operations in the equation.

• Opposite Operations: Used to isolate the variable, such as adding the opposite.
• Simplification: We simplify the equation step-by-step until only the variable is on one side.
• Equality Preservation: A critical aspect of algebraic operations is maintaining equal balance; whatever you do to one side of the equation, you must do to the other.

This method allows us to simplify equations and discover the value of the unknown variable effectively.

One-variable Equations

One-variable equations are equations where we only have to solve for a single unknown. This makes them simpler than multivariable equations. The problem we looked at today is a perfect example, wherein we needed to find the value of \( y \) in the equation \( y - 4 = 7 \).

• Objective: To isolate the variable by employing step-by-step algebraic operations.
• Solution: These equations usually lead to a clear numerical solution, like our example, where \( y \) was found to be 11.
• Types: They can be classified as conditional (like here), identity, or contradiction based on the number and nature of the solutions.

Understanding these equations provides a foundation for more complex algebraic concepts, making them an essential building block in mathematics education.