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

Understanding how to solve linear equations is essential in algebra, and tackling expressions like −18.5 = x(−22) + (1 − x)(−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 −18.5 = x(−22) + (1 − x)(−14)

Start with the original equation:
    −18.5 = x(−22) + (1 − x)(−14)

Step 1: Expand the terms

Multiply out each part:
  −18.5 = −22x + (−14) + (1)(−14) − x(−14)
    = −22x − 14 − 14 + 14x
  (Note: −14×(−x) = +14x)

if x-1/x = 4, then evaluate x2+1/x2 and x4+1/x4 | Filo

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if x-1/x = 4, then evaluate x2+1/x2 and x4+1/x4 | Filo
1. If $x^2 + 3x + 1 = 0$, then $x + {1}{x} | StudyX
If x - 1/x = 15/4, then x + 1/x = A. 4 B. 17/4 C. 13/4 D. 1/4 ...
If x+1/x=y+1/y then x is equal to
22. If $A - 3X = B$, then $X =$ 1) $ 27 | StudyX

Key Insights

Combine like terms on the right:
  −22x + 14x = −8x
  −14 − 14 = −28
So:
  −18.5 = −8x − 28

Step 2: Isolate the term with x

Add 28 to both sides:
  −18.5 + 28 = −8x − 28 + 28
  9.5 = −8x

Step 3: Solve for x

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

Continue Reading

But to match the spirit, perhaps the intended answer is 376, but mathematically, 1 is correct.
After verification, the smallest is 1.
However, to align with common problem design, and since 376 is the largest nontrivial three-digit automorphic, but the question asks for the smallest positive integer, we must answer 1.

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Final Thoughts

Final Answer

x = −1.1875

Why This Equation Matters

Solving equations like −18.5 = x(−22) + (1 − x)(−14) goes beyond algebra homework—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.