Linear equations are the building blocks of algebra. They help us solve real-world problems by turning words into math. We'll learn how to simplify, isolate variables, and find solutions step-by-step.

Understanding different types of linear equations is key. We'll explore conditional equations with specific solutions, identities that are always true, and contradictions with no solution. This knowledge is crucial for tackling more complex math problems.

Solving Linear Equations

Step-by-step linear equation solving

• Simplify each side of the equation separately
  • Combine like terms on each side of the equation (e.g. $3x + 2x = 5x$)
  • Distribute if necessary (e.g. $2(3x - 1) = 6x - 2$)
• Use addition or subtraction to get all variable terms on one side and all constant terms on the other side
  • Add or subtract the same value from both sides to maintain equality (e.g. if $2x + 3 = 7$, then $2x + 3 - 3 = 7 - 3$)
  • Combine like terms after adding or subtracting ($2x = 4$ in the previous example)
• Isolate the variable term by dividing both sides by its coefficient
  • Divide both sides by the same non-zero value to maintain equality (e.g. if $2x = 4$, then $\frac{2x}{2} = \frac{4}{2}$)
  • Simplify the fraction if possible ($x = 2$ in the previous example)
• Check the solution by substituting the value back into the original equation (e.g. if $x = 2$, then $2(2) + 3 = 7$ is true)

Types of linear equations

• Conditional equations
  • Have a solution that makes the equation true
  • Solving the equation yields a specific value for the variable ($x = 2$ for $2x + 3 = 7$)
• Identities
  • True for all values of the variable
  • Solving results in a true statement like $0 = 0$ (e.g. $2(x + 1) = 2x + 2$)
  • Equivalent expressions on both sides of the equals sign
• Contradictions
  • False for all values of the variable
  • Solving results in a false statement like $0 = 1$ (e.g. $2x + 1 = 2x + 2$)
  • Inconsistent equations with no solution

Understanding Linear Equations

• A linear equation is an algebraic expression that forms a straight line when graphed
• Variables are symbols (usually letters) that represent unknown quantities in an equation
• Constants are fixed numerical values in an equation that do not change
• Equation solving involves manipulating the equation to isolate the variable and find its value

Applying Linear Equations

Real-world applications of solutions

• Identify the unknown quantity and assign a variable to represent it (e.g. let $x$ = the number of apples)
• Write an equation that models the situation using the given information
  • Translate verbal phrases into mathematical expressions (e.g. "twice the number of apples" becomes $2x$)
  • Use appropriate units for the variable and constants (e.g. if $x$ represents apples, the equation should not include units of length)
• Solve the equation using the step-by-step approach
• Interpret the solution in the context of the problem
  • Check if the solution makes sense in the given context (e.g. a negative number of apples is not realistic)
  • State the solution using appropriate units (e.g. "The solution is 5 apples.")
• Verify the solution by substituting it back into the original word problem to ensure it satisfies the given conditions