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What is a Linear Equation?

A linear equation is an equation that represents a straight line when graphed on a coordinate plane. It is an algebraic equation in which the highest power of the variable is one. A general form of a linear equation in one variable is ax + b = c, where:

• x is the variable you are solving for.
• a is the coefficient of x, which determines the slope of the line.
• b is the constant, representing the y-intercept (the point where the line crosses the y-axis).
• c is the constant on the right side of the equation.

Linear equations are called “linear” because they describe a line on a graph. The graph of a linear equation is always a straight line, and the equation represents the relationship between two quantities that change at a constant rate. Linear equations are one of the simplest forms of equations and are widely used in various fields like finance, physics, engineering, and many others.

Forms of Linear Equations

There are different forms in which linear equations can be expressed:

• Standard Form: ax + b = c, where a, b, and c are constants.
• Slope-Intercept Form: y = mx + b, where m is the slope and b is the y-intercept.
• Point-Slope Form: y - y1 = m(x - x1), where (x1, y1) is a point on the line and m is the slope.

Steps to Solve a Linear Equation

To solve a linear equation like 2x + 3 = 7, follow these steps:

1. Move the constant term to the right-hand side of the equation by subtracting it from both sides.
2. Isolate the variable x by dividing both sides of the equation by the coefficient of x.
3. The result will give you the value of x, which is the solution to the equation.

For example, in the equation 2x + 3 = 7:

• Step 1: Subtract 3 from both sides: 2x = 7 - 3 → 2x = 4
• Step 2: Divide both sides by 2: x = 4 / 2 → x = 2

Applications of Linear Equations

Linear equations are commonly used in real-world applications to model relationships between variables that change at a constant rate. Some practical applications include:

• Finance: Calculating profit or loss based on sales and expenses.
• Physics: Describing the motion of objects with constant velocity.
• Engineering: Modeling stress and strain in materials.
• Statistics: Performing linear regression to analyze the relationship between variables.

Linear equations form the foundation for understanding more complex algebraic concepts and are widely used to solve problems in everyday life. Whether you are balancing a budget, calculating interest, or analyzing data trends, linear equations are an essential tool in mathematics.