Solving equations with decimals is a crucial skill in algebra. It involves manipulating numbers and variables to find unknown values. This process requires careful attention to decimal arithmetic and the ability to apply inverse operations.

Understanding how to solve decimal equations opens doors to real-world problem-solving. From calculating prices to analyzing scientific data, these skills are essential for making sense of numerical information in everyday life and various professional fields.

Solving Equations with Decimals

Key Components of Algebraic Equations

• Variable: The unknown quantity represented by a letter (e.g., x, y, z)
• Coefficient: The numerical factor of a variable (e.g., in 2x, 2 is the coefficient)
• Solution set: The set of all values that satisfy the equation
• Number line: A visual representation of real numbers, useful for understanding decimal values

Decimal equation verification

• Substitute the given decimal value for the variable in the equation
  • Simplify each side of the equation by performing the necessary arithmetic operations (addition, subtraction, multiplication, division)
  • Compare the left side of the equation to the right side to determine equality
• If the left side equals the right side, the decimal satisfies the equation (verification successful)
• If the left side does not equal the right side, the decimal does not satisfy the equation (verification failed)
  • Example: For the equation $2x + 1.5 = 7.3$, if $x = 2.9$, the equation is satisfied because $2(2.9) + 1.5 = 7.3$

Solving decimal equations

• Perform the same steps as solving equations with integers, but use decimal arithmetic
• Isolate the variable by applying inverse operations to both sides of the equation
  • Add or subtract decimal numbers to both sides of the equation to eliminate decimals on one side (additive inverse)
  • Multiply or divide both sides of the equation by a decimal to eliminate decimals attached to the variable (multiplicative inverse)
• Simplify each side of the equation by performing the necessary arithmetic operations (combine like terms, distribute)
• Check the solution by substituting the decimal value back into the original equation to ensure equality
  • Example: Solve $0.5x - 1.2 = 3.8$
    1. Add 1.2 to both sides: $0.5x = 5$
    2. Multiply both sides by 2: $x = 10$

Word problems with decimals

• Identify the unknown quantity and assign it a variable (x, y, z)
• Translate the word problem into an equation using the given information and the assigned variable
  • Use keywords such as "is," "are," "more than," "less than," "times," and "divided by" to determine the appropriate mathematical operations (equals, addition, subtraction, multiplication, division)
• Solve the equation using the steps for solving equations with decimal numbers
  • Isolate the variable by applying inverse operations to both sides of the equation (additive inverse, multiplicative inverse)
  • Simplify each side of the equation by performing the necessary arithmetic operations (combine like terms, distribute)
• Check the solution by substituting the decimal value back into the word problem to ensure it makes sense in the context of the problem (real-world application)
  • Example: A store sells shirts for $15.99 each. If the total cost of buying x shirts is $79.95, how many shirts were purchased?
    1. Equation: $15.99x = 79.95$
    2. Solve: $x = 5$
    3. Check: $15.99(5) = 79.95$, so 5 shirts were purchased