Linear Equations: SSC CGL Quantitative Aptitude

Formulas (2)

• Solving two equations in two unknowns · Sum and difference of two numbers
  $$x + y = S,\; x - y = D \;\Rightarrow\; x = \frac{S + D}{2},\; y = \frac{S - D}{2}$$
• Solving three equations in three unknowns · Symmetric set of three equations
  $$x + y = p,\; y + z = q,\; z + x = r \;\Rightarrow\; x + y + z = \frac{p + q + r}{2}$$

Watch out for (6)

• Stopping at the first unknown
  → Solving two equations in two unknowns
• Subtracting only part of an equation
  → Solving two equations in two unknowns
• Removing a different unknown from each pair
  → Solving three equations in three unknowns
• Using the same pair twice
  → Solving three equations in three unknowns
• Accepting an option after one equation
  → Testing the options in a system of equations
• Testing the pair before fixing the constant
  → Testing the options in a system of equations

Formulas (1)

• One, none or infinitely many solutions from the coefficient ratios · Number of solutions of two linear equations
  $$\frac{a_1}{a_2} \ne \frac{b_1}{b_2} \;(\text{one}) \qquad \frac{a_1}{a_2} = \frac{b_1}{b_2} \ne \frac{c_1}{c_2} \;(\text{none}) \qquad \frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2} \;(\text{infinitely many})$$

Watch out for (6)

• Mixing forms flips the sign of the constant ratio
  → One, none or infinitely many solutions from the coefficient ratios
• Calling a system with all three ratios different inconsistent
  → One, none or infinitely many solutions from the coefficient ratios
• Choosing "finitely many, but more than one"
  → One, none or infinitely many solutions from the coefficient ratios
• Skipping the constant check for no solution
  → Finding a constant that gives a required number of solutions
• Giving a single value for a unique solution
  → Finding a constant that gives a required number of solutions
• Keeping both roots of the coefficient condition
  → Finding two unknown constants from the ratio conditions

Formulas (4)

• Turning a word problem into one equation in one unknown · Steady change
  $$\text{value after } t \text{ steps} = a + dt \qquad d = \frac{\text{later value} - \text{earlier value}}{\text{steps between them}}$$
• Two unknowns from two statements · Equal after a transfer of k
  $$A - k = B + k \;\Rightarrow\; A - B = 2k$$
• Swapped quantities and the change in the bill · Swapped quantities
  $$rq_1 + q_2 \;\to\; rq_2 + q_1 \qquad (a + b)(x + y) = P + Q \qquad (a - b)(x - y) = P - Q$$
• Whole-number answers: counting coins and notes · Coins of ₹5, ₹2 and ₹1
  $$x + y + z = N,\quad 5x + 2y + z = V \;\Rightarrow\; 4x + y = V - N$$

Watch out for (8)

• Using two letters where one will do
  → Turning a word problem into one equation in one unknown
• Miscounting the steps between two values
  → Turning a word problem into one equation in one unknown
• Answering x instead of the question
  → Turning a word problem into one equation in one unknown
• Taking the transfer as the difference
  → Two unknowns from two statements
• Comparing totals over different numbers of days
  → Two unknowns from two statements
• Swapping the prices instead of the quantities
  → Swapped quantities and the change in the bill
• Taking the percentage of the new bill
  → Swapped quantities and the change in the bill
• Checking only one count for a negative value
  → Whole-number answers: counting coins and notes

Formulas (1)

• The gap between two ages never changes · Age-gap shortcut
  $$\text{younger then} = \frac{\text{gap}}{k - 1} \qquad \text{one ratio part} = \frac{\text{gap}}{m - n}$$

Watch out for (6)

• Moving only one person in time
  → The gap between two ages never changes
• Expecting the ratio of ages to stay fixed
  → The gap between two ages never changes
• Using today's ages inside a past condition
  → Age conditions at two different times
• Reading "times more than" one way only
  → Age conditions at two different times
• Stopping at one age when the question asks for a combination
  → Age conditions at two different times
• Deciding who is older when the question does not say
  → Three or more people linked by relations