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Aptitude Topics

Linear Equations

A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and a single variable. The graph of a linear equation is always a straight line.

Fundamental Principles

Linear Equation in One Variable

An equation of the form ax + b = 0, where a ≠ 0, with exactly one solution: x = -b/a.

Linear Equation in Two Variables

An equation of the form ax + by + c = 0, where a and b are not both zero. It has infinitely many solutions forming a straight line.

System of Linear Equations

A set of two or more linear equations with the same variables. The solution is the point (or points) that satisfies all equations simultaneously.

Essential Formulation Tips

  • Isolate the variable by performing the same operation on both sides.
  • For two-variable systems, use substitution when one variable is easily expressed in terms of the other.
  • Use elimination when coefficients can be made equal by multiplication.
  • Always verify your solution by substituting back into the original equation.

Shortcut Execution Techniques

  • Cross-multiplication: a/b = c/d → ad = bc.
  • For ax + b = cx + d, group x terms: (a-c)x = d-b.
  • In word problems, assign the unknown to the quantity being asked.
  • For simultaneous equations, multiply to make one coefficient equal before adding/subtracting.

Contextual Inquiries (FAQs)

Q: How many solutions does a linear equation in one variable have?

A: Exactly one solution (unless it simplifies to a contradiction like 0 = 1, giving no solution, or 0 = 0, giving infinitely many).

Q: What is the substitution method?

A: Solve one equation for one variable, then substitute that expression into the other equation.

Q: When do two lines have no solution?

A: When they are parallel — same slope but different y-intercepts.