Introduction to Linear Equations

Introduction to Linear Equations

• A linear equation is an equation that can be written in the form ax + b = 0, where 'a' and 'b' are constants, and 'x' is a variable.
• Linear equations represent a straight line on a graph.
• Solving a linear equation involves finding the value of the variable that makes the equation true.

Key Concepts

• Variable: A symbol (usually a letter) that represents an unknown numerical value.
• Constant: A fixed numerical value.
• Coefficient: The numerical factor that multiplies a variable.
• Solution: The value of the variable that satisfies the equation.

Methods for Solving Linear Equations

• Addition/Subtraction method: Add or subtract the same value to both sides of the equation to isolate the variable term.
• Example: 2x + 5 = 11, subtract 5 from both sides to get: 2x = 6.
• Multiplication/Division method: Multiply or divide both sides of the equation by the same value (not zero) to isolate the variable.
• Example: (x/3) = 4, multiply both sides by 3 to get: x = 12.
• Combining like terms: Simplify each side of the equation by combining like terms (variables with the same power).
• Example: 3x + 2x - 5 = 10 becomes 5x - 5 = 10.

Solving Equations with Multiple Steps

• Many equations require more than one step to solve.
• Work through each step, keeping the equation balanced.
• Follow the order of operations (PEMDAS/BODMAS), but in reverse when isolating the variable.
• Example: 2(x-3) + 4 = 10
  • Distribute: 2x - 6 + 4 = 10
  • Simplify: 2x - 2 = 10
  • Add 2 to both sides: 2x = 12
  • Divide by 2: x = 6

Solving Equations with Fractions

• Multiply both sides of the equation by the least common denominator (LCD) to eliminate fractions.
• Example: (x/2) + 3 = 7
  • Multiply by 2: x + 6 = 14
  • Subtract 6: x = 8

Solving Equations with Parentheses

• Distribute the number outside the parentheses to the terms inside.
• Treat the expression inside the parenthesis as a single unit before isolating the variable.
• Example: 2(x + 3) = 10
  • Distribute: 2x + 6 = 10
  • Subtract 6: 2x = 4
  • Divide by 2: x = 2

Identifying Solutions

• A solution to an equation is a value that makes the equation true.
• Substitute the potential solution back into the original equation to verify. If the equation holds true, then your solution is correct.

Solving Equations with Variables on Both Sides

• Move all the variable terms to one side of the equation and all the constant terms to the other side.
• Example: 3x + 5 = x + 9
  • Subtract x from both sides: 2x + 5 = 9
  • Subtract 5 from both sides: 2x = 4
  • Divide by 2: x = 2

Solving Equations with No Solutions or Infinitely Many Solutions

• Some equations have no solution (inconsistent equations); they produce a false statement.
• Some equations have infinitely many solutions (dependent equations); they produce an identity. Recognize these situations when simplifying.

Word Problems

• Translate word problems into equations.
• Define the variables.
• Use the information given to create an equation.
• Solve the equation.
• Interpret your solution in the context of the word problem.

Example Problem

• If 3 times a number, plus 7, is equal to 16, what is the number?
  • Let x = the number
  • Equation: 3x + 7 = 16
  • Solving the equation: 3x = 9, x = 3.

This quiz covers the fundamentals of linear equations, including their definition, key concepts, and methods for solving them. Explore the principles that govern solving equations of the form ax + b = 0, and learn various techniques such as addition, subtraction, multiplication, and division methods.