Eduqo
Quantitative Aptitude›Ch 4: Algebra, Geometry & Mensuration›Linear Equations
Master one and two variable equations, consistency rules, and word problems for competitive exams.
To solve any system of linear equations uniquely, you need at least as many independent equations as there are variables. Always check consistency ratios before spending time solving two-variable simultaneous equations.
Memory Hooks
| Condition | Graphical View | Solution Type |
|---|---|---|
| a1/a2 ≠ b1/b2 | Intersecting | Unique |
| a1/a2 = b1/b2 ≠ c1/c2 | Parallel | None |
| a1/a2 = b1/b2 = c1/c2 | Coincident | Infinite |
ax + b = 0
(Single Variable)
Single Variable
Straight number line point solution.
ax + by = c
(Two Variables)
Two Variables
Represents a straight line on a graph.
a1/a2 ≠ b1/b2
(Single Cross)
Single Cross
Lines cross at exactly one point.
Word Problem
(Translation)
Translation
Convert statements into math equations.
Assuming 1 equation solves 2 variablesNeed 2 independent equations for 2 variables
Without a second equation, infinite value pairs exist.
Forgetting constant sign in ratiosAlways bring all terms to left: ax + by + c = 0
Sign errors in c1 and c2 wreck parallel/coincident checks.
Treating dependent equations as uniqueCheck if second equation is a multiple of first
2x+3y=5 and 4x+6y=10 give infinite solutions, not one.
Q1.For what value of k will the equations 3x + y = 1 and (2k - 1)x + (k - 1)y = 2k + 1 have no solution?
Q2.The sum of numerator and denominator of a fraction is 12. If 3 is added to the numerator, the fraction becomes 1/2. Find the fraction.