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 balance equations, simplify complex ones, and tackle word problems step-by-step.

Checking solutions is crucial to ensure our answers make sense. We'll practice substituting our results back into equations and breaking down tricky problems. These skills will set us up for success in more advanced math topics.

Solving Linear Equations

Properties of equality in linear equations

• Subtraction Property of Equality subtracts the same quantity from both sides of an equation, maintaining equality
  • $a = b$, then $a - c = b - c$ (subtracting $c$ from both sides)
• Addition Property of Equality adds the same quantity to both sides of an equation, maintaining equality
  • $a = b$, then $a + c = b + c$ (adding $c$ to both sides)
• Isolate the variable on one side by applying Subtraction or Addition Property of Equality
  • Subtract quantity from both sides if added to variable ($x + 3 = 7$ becomes $x = 7 - 3$)
  • Add quantity to both sides if subtracted from variable ($x - 3 = 7$ becomes $x = 7 + 3$)
• Substitute solution back into original equation to check accuracy ($x = 4$ in $x + 3 = 7$ yields $4 + 3 = 7$, which is true)
• These properties maintain the balance of the equation, ensuring equality is preserved

Simplification of complex equations

• Combine like terms on each side of equation
  • Like terms have same variables with same exponents ($3x$ and $5x$ are like terms, $3x$ and $5x^2$ are not)
• Simplify each side by performing indicated operations in order of operations
  1. Parentheses
  2. Exponents
  3. Multiplication and division (left to right)
  4. Addition and subtraction (left to right)
• Isolate variable using Subtraction and Addition Properties of Equality
• Solve for variable and check solution ($2x + 3 = 5x - 7$ becomes $-3x = -10$, so $x = \frac{10}{3}$)

Word problems to mathematical equations

• Identify unknown quantity and assign variable ($x$ often represents unknown)
• Translate word problem into equation using given information
  • Key phrases indicate operations ("sum" for addition, "difference" for subtraction, "product" for multiplication, "quotient" for division)
• Solve equation using Subtraction and Addition Properties of Equality
• Interpret solution in context of word problem ("The number is 5" instead of just "5")

Real-world applications of linear equations

• Identify given information and question being asked
• Assign variable to unknown quantity
• Write equation representing relationship between known and unknown quantities
• Solve equation using Subtraction and Addition Properties of Equality
• Check if solution makes sense in problem context
• State answer in complete sentence ("The length of the rectangle is 12 cm" instead of just "12")

Checking Solutions and Problem-Solving Strategies

Check solutions by substituting the result back into the original equation

• Replace variable in original equation with obtained solution
• Simplify both sides of equation
• Solution is correct if left side equals right side ($x = 4$ in $2x + 3 = 11$ yields $2(4) + 3 = 11$, which simplifies to $11 = 11$, confirming the solution)
• The set of all values that satisfy the equation is called the solution set

Apply problem-solving strategies to tackle challenging equations and word problems

• Break down complex problems into smaller, manageable steps
• Identify key information and question being asked
• Use logical reasoning and properties of equality to solve problem
• Check solution and ensure it makes sense in problem context
• Try alternative approaches or seek guidance from examples or resources if stuck

Understanding Algebraic Expressions and Equations

• An algebraic expression is a combination of variables, numbers, and operations without an equals sign
• An equation is a mathematical statement that two expressions are equal, indicated by an equals sign
• Solving an equation involves finding the value(s) that make the equation true