Introduction
Linear equations are an essential part of algebra and play a significant role in problem-solving and real-life scenarios. Through detailed explanations and numerous examples, you’ll gain a solid grasp of linear equations, their solutions, and their graphical representations.
A linear equation is an equation that can be written in the form of ax + b = c, where a, b, and c are real numbers, and a ≠ 0. The variable x is called the unknown, and the coefficients a and b are called the constants.
Types of linear equations
There are two main types of linear equations:
One-variable linear equations: These equations have one variable, and they can be solved by isolating the variable on one side of the equation. Ex: 2x+5=0 is a linear equation with one variable.
Two-variable linear equations: These equations have two variables, and they can be solved by graphing the two sides of the equation on a coordinate plane. The solution to the equation is the point where the two lines intersect. Ex: 5x+y=0 is a linear equation with two variables.
Solving linear equations
There are many different ways to solve linear equations. Some common methods include:
Combining like terms: Like terms are terms that have the same variables and the same powers of those variables. For example, the terms 3x and 2x are like terms. Combining like terms involves adding or subtracting the coefficients of the like terms.
Solving for x: Solving for x involves isolating x on one side of the equation. This can be done by adding or subtracting terms, multiplying or dividing both sides of the equation by a constant, or using the distributive property.
Using the graphical method: The graphical method involves graphing the two sides of the equation on a coordinate plane. The solution to the equation is the point where the two lines intersect.
Linear equations in real-world problems
Linear equations are used in many real-world problems. For example, we might use linear equations to represent the cost of a product, the distance traveled by a car, or the amount of money in a savings account. We might also use linear equations to represent the relationship between two variables, such as the height and weight of a person or the price and demand for a product.
Examples
Here are some examples of linear equations that students might encounter in class 8:
3x + 2 = 10
x – 5 = 0
2x + 3y = 6
y = 2x – 1
Applications of linear equations
Linear equations are used in many different applications, such as:
Physics: Linear equations are used to model the motion of objects.
Chemistry: Linear equations are used to model chemical reactions.
Economics: Linear equations are used to model economic relationships.
Engineering: Linear equations are used to design and analyze structures.
I hope this guide helps you understand linear equations for class 8!