Solving the Equation: -18.5 = x(-22) + (1 - x)(-14)

Understanding how to solve linear equations is essential in algebra, and tackling expressions like βˆ’18.5 = x(βˆ’22) + (1 βˆ’ x)(βˆ’14) reveals key techniques that simplify complex formulas into clear, solvable forms. Whether you're a student mastering algebra or a teacher explaining key concepts, this step-by-step guide will help you solve the equation efficiently and boost your problem-solving skills.

Understanding the Context

Step-by-Step Solution to βˆ’18.5 = x(βˆ’22) + (1 βˆ’ x)(βˆ’14)

Start with the original equation:
β€ƒβ€ƒβ€ƒβ€ƒβˆ’18.5 = x(βˆ’22) + (1 βˆ’ x)(βˆ’14)

Step 1: Expand the terms

Multiply out each part:
β€ƒβ€ƒβˆ’18.5 = βˆ’22x + (βˆ’14) + (1)(βˆ’14) βˆ’ x(βˆ’14)
    = βˆ’22x βˆ’ 14 βˆ’ 14 + 14x
  (Note: βˆ’14Γ—(βˆ’x) = +14x)

Then: -18.5 = x(-22) + (1 - x)(-14)

Key Insights

Combine like terms on the right:
β€ƒβ€ƒβˆ’22x + 14x = βˆ’8x
β€ƒβ€ƒβˆ’14 βˆ’ 14 = βˆ’28
So:
β€ƒβ€ƒβˆ’18.5 = βˆ’8x βˆ’ 28

Step 2: Isolate the term with x

Add 28 to both sides:
β€ƒβ€ƒβˆ’18.5 + 28 = βˆ’8x βˆ’ 28 + 28
  9.5 = βˆ’8x

Step 3: Solve for x

Divide both sides by βˆ’8:
  x = 9.5 / (βˆ’8)
  x = βˆ’1.1875

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Final Thoughts

Final Answer

x = βˆ’1.1875

Why This Equation Matters

Solving equations like βˆ’18.5 = x(βˆ’22) + (1 βˆ’ x)(βˆ’14) goes beyond algebra homeworkβ€”it’s foundational for modeling real-world scenarios such as budgeting, proportional reasoning, and physics equations. Mastering this process strengthens logical thinking and prepares you for advanced math topics like systems of equations, quadratic functions, and calculus.

Tips for Solving Similar Equations

  • Always expand parentheses clearly.
  • Combine like terms before solving for the variable.
  • Isolate the variable term step-by-step.
  • Check your solution by substituting back into the original equation.