Pair of Linear Equations in Two Variables — Practice

Question 1

What is the condition for a pair of linear equations to have NO solution?

Accepted Answer

✦ Answer: a₁/a₂ = b₁/b₂ ≠ c₁/c₂. The lines are parallel and never intersect.

Question 2

Solve: x + y = 5 and x − y = 1.

Accepted Answer

✦ Answer: Adding both: 2x = 6, x = 3. Substituting: y = 2. Solution: (3, 2).

Question 3

Solve by elimination: 2x + 3y = 7 and 4x − y = 5.

Accepted Answer

Step 1 — Make y coefficients opposite.
  Multiply second equation by 3: 12x − 3y = 15

Step 2 — Add to first equation.
  (2x + 3y) + (12x − 3y) = 7 + 15
  14x = 22
  x = 22/14 = 11/7

Step 3 — Substitute to find y.
  2(11/7) + 3y = 7
  22/7 + 3y = 7
  3y = 7 − 22/7 = 49/7 − 22/7 = 27/7
  y = 9/7

Step 4 — Verify.
  Check both equations.

✦ Answer: x = 11/7, y = 9/7.

Question 4

A boat travels 30 km downstream in 2 hours and 18 km upstream in 2 hours. Find the speed of the boat in still water and the speed of the stream.

Accepted Answer

Step 1 — Set up variables.
  Let boat speed in still water = x km/h, stream speed = y km/h.

Step 2 — Form equations.
  Downstream speed = x + y; distance/time: 30/(x+y) = 2 ⟹ x + y = 15
  Upstream speed = x − y; distance/time: 18/(x−y) = 2 ⟹ x − y = 9

Step 3 — Solve by elimination.
  Add: (x+y) + (x−y) = 15 + 9
  2x = 24 ⟹ x = 12
  Substitute: 12 + y = 15 ⟹ y = 3

Step 4 — Verify.
  Downstream: 12+3 = 15 km/h; 30/15 = 2 hours ✓
  Upstream: 12−3 = 9 km/h; 18/9 = 2 hours ✓

✦ Answer: Boat speed in still water = 12 km/h; Stream speed = 3 km/h.

Practice — Pair of Linear Equations in Two Variables

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