Linear vs. Non-Linear Equations: Definitions, Examples, and Solutions

Understand the difference between linear and non-linear equations. This guide defines linear equations, explains their properties, and contrasts them with non-linear equations using examples and step-by-step solutions.

Linear vs. Non-Linear Equations

What is an Equation?

An equation is a mathematical statement asserting that two expressions are equal. It contains an equals sign (=), separating a left-hand side (LHS) and a right-hand side (RHS). Equations can involve numbers, variables, or both.

Linear Equations

A linear equation is an equation where the highest power of any variable is 1. When graphed, a linear equation always forms a straight line. This means there's a constant rate of change between the variables.

Forms of Linear Equations

• One Variable: ax + b = c (a is the coefficient, b and c are constants, x is the variable)
• Two Variables: ax + by = c (a and b are coefficients, c is a constant, x and y are variables)

Examples: Solving Linear Equations

Example 1: One-Variable Equation

(The solution to the equation 9(x+1) = 2(3x+8) is provided in the original text and should be included here, showing the step-by-step solution.)

Example 2: System of Two Equations (Two Variables)

(The solution to the equations x + 2y = 1 and x = y is provided in the original text and should be included here, clearly showing the substitution method used to find the values of x and y.)

Example 3: Simple One-Variable Equation

(The solution to the equation 3x + 9 = 2x + 18 is provided in the original text and should be included here.)

Example 4: Word Problem Leading to a One-Variable Equation

(The word problem about ten times a number equaling 50 is provided in the original text. The formulation of the linear equation (10x = 50) and its solution should be given here.)

Example 5: Word Problem Leading to a System of Two Equations

(The word problem involving the sum of two numbers is given in the original text. The formulation of the two linear equations and their solution to find the two numbers should be given here.)

Example 6: Solving a Simple Linear Equation

(The solution to the equation 7x + 21 = 6x + 26 is provided in the original text and should be included here.)

Example 7: Solving a One-Variable Equation

(The solution to the equation x = 12(x+2) is provided in the original text and should be included here.)

Non-Linear Equations

A non-linear equation is any equation that is not linear. This means it has variables raised to powers other than 1 (e.g., x², x³, √x) or involves trigonometric functions, exponential functions, logarithmic functions, etc. The graph of a non-linear equation is a curve, not a straight line.

Examples: Non-Linear Equations

Example 1: Solving a Quadratic Equation

(The solution to the quadratic equation 3x² - 5x + 2 = 0 is provided in the original text and should be included here, showing the factorization method used to find the solutions.)

Differences Between Linear and Non-Linear Equations

Conclusion

Linear and non-linear equations are fundamental in mathematics. Understanding their differences and how to solve them is essential for solving various types of problems.

Solving Linear Equations: A Simple Example

Let's solve the linear equation 7x + 21 = 6x + 26. Our goal is to find the value of x that makes this equation true.

Solution

1. Rearrange the equation: We want to get all the terms with 'x' on one side of the equation and all the constant terms on the other side. We can do this by subtracting 6x from both sides and subtracting 21 from both sides.
2. Simplify: This gives us 7x - 6x = 26 - 21.
3. Solve for x: This simplifies to x = 5.

Therefore, the solution to the equation 7x + 21 = 6x + 26 is x = 5. You can check this by substituting x = 5 back into the original equation: 7(5) + 21 = 56 and 6(5) + 26 = 56. Both sides are equal, confirming our solution is correct.