2.3 Solving Equations Using the Subtraction and Addition Properties of Equality

Solving equations is all about balance. By using the subtraction and addition properties of equality, we can isolate variables and find their values. These properties let us add or subtract the same amount from both sides of an equation, keeping things equal.

Real-world problems often involve equations. We can translate word problems into algebraic equations, then solve them using these properties. This skill helps us tackle everyday math challenges, from figuring out prices to calculating distances.

Solving Equations Using the Subtraction and Addition Properties of Equality

Equation verification process

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• Substitute the given number for the variable in the equation (e.g., if $x = 3$, replace all instances of $x$ with 3)
• Simplify the left side of the equation by performing any necessary operations (addition, subtraction, multiplication, division)
• Simplify the right side of the equation by performing any necessary operations (addition, subtraction, multiplication, division)
• Compare the left and right sides of the equation
  • If the left side equals the right side, the number satisfies the equation (e.g., $3 + 2 = 5$ is true, so $x = 3$ satisfies the equation $x + 2 = 5$)
  • If the left side does not equal the right side, the number does not satisfy the equation (e.g., $3 + 2 ≠ 6$, so $x = 3$ does not satisfy the equation $x + 2 = 6$)
• This process helps maintain balanced equations, ensuring both sides remain equal

Subtraction property in equations

• The Subtraction Property of Equality states that if you subtract the same quantity from both sides of an equation, the equation remains true (e.g., if $x + 3 = 7$, then $x + 3 - 3 = 7 - 3$, which simplifies to $x = 4$)
• To solve an equation using the Subtraction Property of Equality:
  1. Identify the term you want to isolate on one side of the equation (usually the variable term)
  2. Subtract the unwanted term from both sides of the equation (e.g., if the equation is $x + 5 = 9$, subtract 5 from both sides)
  3. Simplify both sides of the equation (e.g., $x + 5 - 5 = 9 - 5$ simplifies to $x = 4$)
  4. The variable is now isolated, and the equation is solved

Addition property in equations

• The Addition Property of Equality states that if you add the same quantity to both sides of an equation, the equation remains true (e.g., if $x - 2 = 6$, then $x - 2 + 2 = 6 + 2$, which simplifies to $x = 8$)
• To solve an equation using the Addition Property of Equality:
  1. Identify the term you want to isolate on one side of the equation (usually the variable term)
  2. Add the unwanted term to both sides of the equation (e.g., if the equation is $x - 3 = 5$, add 3 to both sides)
  3. Simplify both sides of the equation (e.g., $x - 3 + 3 = 5 + 3$ simplifies to $x = 8$)
  4. The variable is now isolated, and the equation is solved

Word problems to algebraic equations

• Identify the unknown quantity and represent it with a variable (e.g., let $x$ represent the number of apples)
• Translate the given information into mathematical expressions (algebraic expressions)
  • Use keywords to determine the appropriate operations (e.g., "sum" for addition, "difference" for subtraction, "product" for multiplication, "quotient" for division)
• Set up an equation by equating two expressions (e.g., if the problem states "the sum of a number and 5 is 12," the equation would be $x + 5 = 12$)
• Solve the equation using the appropriate properties of equality (Subtraction or Addition Property)

Real-world equation applications

• Read the problem carefully and identify the given information and the unknown quantity (e.g., the problem might give the total cost and the number of items purchased, and ask for the price per item)
• Assign a variable to represent the unknown quantity (e.g., let $x$ represent the price per item)
• Write an equation that represents the relationship between the given information and the unknown quantity (e.g., if the total cost is $30 for 5 items, the equation would be$5x = 30$)
• Solve the equation using the Subtraction or Addition Property of Equality (e.g., divide both sides by 5 to get $x = 6$)
• Interpret the solution in the context of the original problem (e.g., the price per item is $6)
• Check the solution by substituting it back into the original equation or problem (e.g., $5 ×$6 = $30, which matches the given total cost)

Addition vs subtraction in equations

• Use the Addition Property of Equality when the term you want to isolate has a negative coefficient (e.g., in the equation $-2x + 3 = 7$, add 2x to both sides to isolate $x$)
• Use the Subtraction Property of Equality when the term you want to isolate has a positive coefficient (e.g., in the equation $3x - 4 = 8$, subtract 3x from both sides to isolate $x$)
• Consider the sign of the term you want to eliminate when deciding which property to use (e.g., if the term is positive, use subtraction; if the term is negative, use addition)
• Remember that the goal is to isolate the variable term on one side of the equation by eliminating the unwanted terms

Mathematical Properties and Equation Solving Strategies

• Understand and apply mathematical properties such as the commutative, associative, and distributive properties to simplify equations
• Develop algebraic reasoning skills to analyze and solve more complex equations
• Use various equation solving strategies, including combining like terms and using inverse operations, to efficiently solve equations