Solving equations is a fundamental skill in algebra. It involves using properties of equality to isolate variables and find their values. This process is crucial for understanding more complex mathematical concepts and solving real-world problems.

The addition and subtraction properties of equality are key tools for solving linear equations. These properties allow us to manipulate equations while maintaining balance, helping us isolate variables and find solutions efficiently.

Solving Equations Using Subtraction and Addition Properties of Equality

Verification of linear equation solutions

• Understand the concept of a solution to an equation
  • Value that makes the equation true when substituted for the variable ($x = 3$ in $2x + 1 = 7$)
• Substitute the given value for the variable in the original equation
• Simplify the left side of the equation by performing operations (addition, subtraction, multiplication, division)
• Simplify the right side of the equation by performing operations
• Check if the left side equals the right side
  • Equal sides confirm the given value is a solution ($6 + 1 = 7$)
  • Unequal sides indicate the given value is not a solution ($6 + 1 ≠ 8$)

Properties for equation solving

• Addition Property of Equality
  • Adding the same value to both sides of an equation maintains equality ($a = b$ implies $a + c = b + c$)
  • Example: If $x + 3 = 7$, then $x + 3 - 3 = 7 - 3$ and $x = 4$
• Subtraction Property of Equality
  • Subtracting the same value from both sides of an equation maintains equality ($a = b$ implies $a - c = b - c$)
  • Example: If $x - 5 = 2$, then $x - 5 + 5 = 2 + 5$ and $x = 7$
• Isolate the variable term on one side of the equation
  • Add or subtract the same value on both sides to eliminate the constant term on the variable side ($x + 3 = 7$ becomes $x = 7 - 3$)
• Simplify both sides of the equation ($x = 4$)
• The value on the side without the variable is the solution

Equations with simplification steps

• Simplify each side of the equation by combining like terms ($2x + 3x - 4 = 5$ becomes $5x - 4 = 5$)
• Use the Distributive Property to remove parentheses, if necessary
  • $a(b + c) = ab + ac$ ($2(x + 3) = 10$ becomes $2x + 6 = 10$)
• Isolate the variable term on one side of the equation
  • Add or subtract the same value on both sides to eliminate the constant term on the variable side ($5x - 4 = 5$ becomes $5x = 9$)
• Simplify both sides of the equation ($x = \frac{9}{5}$)
• The value on the side without the variable is the solution

Word problems to equations

• Identify the unknown quantity and assign a variable to it (let $x$ represent the unknown number)
• Translate the word problem into an equation using the given information
  • Use key phrases to determine the operations and relationships between quantities
    • "sum," "more than," or "increased by" indicate addition ($x + 5 = 12$)
    • "difference," "less than," or "decreased by" indicate subtraction ($x - 3 = 7$)
    • "product," "times," or "multiplied by" indicate multiplication ($2x = 18$)
    • "quotient," "divided by," or "ratio" indicate division ($\frac{x}{4} = 6$)
• Write an equation that represents the problem ("5 more than an unknown number is 12" becomes $x + 5 = 12$)
• Solve the equation using the appropriate properties of equality ($x = 7$)

Real-world applications of linear equations

• Read the problem carefully and identify the given information and the question to be answered
• Assign a variable to the unknown quantity (let $x$ represent the number of hours worked)
• Write an equation that represents the problem using the given information ("total pay is $15 per hour plus a $50 bonus" becomes $15x + 50 = 290$)
• Solve the equation using the subtraction and addition properties of equality ($x = 16$)
• Interpret the solution in the context of the original problem (16 hours were worked)
• Check if the solution makes sense in the given context
  • If not, review the problem-solving steps for errors (negative hours worked is not possible)

Fundamental Concepts in Equation Solving

• Algebraic expressions: Combinations of variables, numbers, and operations (e.g., 2x + 3)
• Equation solving: The process of finding the value(s) of a variable that make an equation true
• Mathematical reasoning: Logical thinking used to analyze and solve mathematical problems
• Number sense: Understanding of numbers, their relationships, and operations
• Algebraic manipulation: Rearranging and simplifying algebraic expressions to solve equations