Summary of First Degree Equations

Goals

1. Understand and solve first-degree polynomial equations in the form ax + b = c using the properties of equality.

2. Apply knowledge of first-degree equations to real-life scenarios, such as calculating taxi fares.

Contextualization

First-degree equations are a common part of our everyday experiences. For instance, when you are planning a taxi ride, you need to consider a base fare along with a per-kilometer charge. By understanding how to solve these equations, you can easily calculate the total fare for your journey. For example, if the base fare is ₹450 and the charge per kilometer is ₹120, you can easily determine the overall cost based on how far you intend to travel.

Subject Relevance

To Remember!

Definition of a First-Degree Equation

A first-degree equation is a mathematical expression represented in the form ax + b = c, where 'a', 'b', and 'c' are constants and 'x' is the variable. The defining aspect of these equations is that the variable is only raised to the first power.

• First-degree equations are linear.

• The solution indicates the value of 'x' that satisfies the equation.

• These equations are essential for solving problems involving ratios and costs.

Properties of Equality

The properties of equality are mathematical principles that enable you to manipulate equations to isolate the variable. The core properties are addition, subtraction, multiplication, and division of both sides of the equation by the same number.

• Addition: If a = b, then a + c = b + c.

• Subtraction: If a = b, then a - c = b - c.

• Multiplication: If a = b, then a * c = b * c (as long as c ≠ 0).

• Division: If a = b, then a / c = b / c (as long as c ≠ 0).

Solving First-Degree Equations in the Form ax + b = c

To solve a first-degree equation, the goal is to isolate the variable 'x' on one side of the equation. This process typically involves applying the properties of equality to simplify the expression.

• Isolating 'x' often requires several steps.

• First, eliminate constant terms from the variable's side.

• Next, either divide or multiply as necessary to find 'x'.

Practical Applications

• Calculating taxi fares based on a fixed charge and variable charge per kilometer traveled.

• Personal financial management, such as budgeting with varying incomes and expenses.

• Evaluating service contracts like telecoms and internet plans, which define costs using fixed and variable rates.

Key Terms

• First-Degree Equation: A linear equation in the format ax + b = c.

• Properties of Equality: Mathematical principles for manipulating equations.

• Fixed Rate: A fee that remains constant regardless of usage.

• Variable Rate: A fee that adjusts in proportion to usage.

Questions for Reflections

• In what ways can first-degree equations be applied in everyday situations, like planning a trip?

• What challenges might arise when solving first-degree equations, and how can you address them?

• How does understanding the properties of equality enhance your ability to solve mathematical problems adeptly?

Planning a Taxi Trip

Utilize your understanding of first-degree equations to estimate the total fare for a taxi ride over various distances.

Instructions

• Assume the base fare for a taxi ride is ₹450 and the charge is ₹120 for each kilometer traveled.

• Formulate the equation that represents the total cost (c) for traveling x kilometers.

• Solve the equation to find the total fare for distances of 5 km, 10 km, and 20 km.

• Compare the fares and consider how minor changes in the distance can affect the total ride cost.