Pair of Linear Equations in Two Variables — Practice

Question 1

For what value of k do the equations 2x + 3y = 4 and (k+2)x + 6y = 3k+2 represent intersecting lines? State the condition.

Accepted Answer

Step 1 — Intersecting lines need a₁/a₂ ≠ b₁/b₂.

Step 2 — 2/(k+2) ≠ 3/6 = 1/2 ⇒ (k+2) ≠ 4 ⇒ k ≠ 2.

✦ Answer: any k ≠ 2.

Question 2

Solve x + y = 14 and x − y = 4.

Accepted Answer

Step 1 — Add: 2x = 18 ⇒ x = 9.

Step 2 — y = 14 − 9 = 5.

✦ Answer: x = 9, y = 5.

Question 3

Are the lines 3x + 2y = 5 and 6x + 4y = 9 consistent?

Accepted Answer

Step 1 — 3/6 = 2/4 = 1/2 but 5/9 ≠ 1/2, so a₁/a₂ = b₁/b₂ ≠ c₁/c₂.

Step 2 — Parallel lines ⇒ no solution.

✦ Answer: Inconsistent (no solution).

Question 4

Solve 2x + 3y = 11 and 2x − 4y = −24 by elimination.

Accepted Answer

Step 1 — Subtract: (3y − (−4y)) = 11 − (−24) ⇒ 7y = 35 ⇒ y = 5.

Step 2 — 2x + 15 = 11 ⇒ x = −2.

✦ Answer: x = −2, y = 5.

Question 5

Solve s − t = 3 and s/3 + t/2 = 6 by substitution.

Accepted Answer

Step 1 — s = t + 3.

Step 2 — (t+3)/3 + t/2 = 6 ⇒ multiply by 6: 2(t+3) + 3t = 36 ⇒ 5t + 6 = 36 ⇒ t = 6.

Step 3 — s = 9.

✦ Answer: s = 9, t = 6.

Question 6

Solve 2/x + 3/y = 13 and 5/x − 4/y = −2.

Accepted Answer

Step 1 — Let u = 1/x, v = 1/y: 2u + 3v = 13 and 5u − 4v = −2.

Step 2 — Multiply: (×4) 8u + 12v = 52; (×3) 15u − 12v = −6. Add: 23u = 46 ⇒ u = 2.

Step 3 — 2(2) + 3v = 13 ⇒ v = 3.

Step 4 — x = 1/u = 1/2, y = 1/v = 1/3.

✦ Answer: x = 1/2, y = 1/3.

Question 7

Find k so that x + 2y = 3 and 5x + ky + 15 = 0 have infinitely many solutions.

Accepted Answer

Step 1 — Write both as = 0: x + 2y − 3 = 0 and 5x + ky + 15 = 0.

Step 2 — Need 1/5 = 2/k = −3/15 = −1/5.

Step 3 — But 1/5 ≠ −1/5, so there is NO value of k giving infinite solutions.

✦ Answer: no such k exists (the constant ratio contradicts the coefficient ratio).

Question 8

The sum of a two-digit number and the number formed by reversing its digits is 66. The digits differ by 2. Find the number(s).

Accepted Answer

Step 1 — Let tens digit x, units y: (10x+y)+(10y+x)=66 ⇒ 11(x+y)=66 ⇒ x+y=6.

Step 2 — Digits differ by 2: x − y = ±2.

Step 3 — x+y=6, x−y=2 ⇒ x=4, y=2 (number 42); x−y=−2 ⇒ x=2, y=4 (number 24).

✦ Answer: 42 or 24.

Question 9

A boat goes 30 km upstream and 44 km downstream in 10 hours. It goes 40 km upstream and 55 km downstream in 13 hours. Find the speed of the boat in still water and of the stream.

Accepted Answer

Step 1 — Let boat speed x, stream y: upstream (x−y), downstream (x+y). Put u = 1/(x−y), v = 1/(x+y).

Step 2 — 30u + 44v = 10 and 40u + 55v = 13.

Step 3 — Solve: (×4) 120u+176v=40; (×3) 120u+165v=39 ⇒ 11v = 1 ⇒ v = 1/11 ⇒ x+y = 11.

Step 4 — 30u + 44/11 = 10 ⇒ 30u = 6 ⇒ u = 1/5 ⇒ x−y = 5.

Step 5 — x+y=11, x−y=5 ⇒ x=8, y=3.

✦ Answer: boat 8 km/h, stream 3 km/h.

Question 10

Five years ago Nuri was thrice as old as Sonu. Ten years later Nuri will be twice as old as Sonu. Find their present ages.

Accepted Answer

Step 1 — Let present ages Nuri x, Sonu y: (x−5) = 3(y−5) ⇒ x − 3y = −10.

Step 2 — (x+10) = 2(y+10) ⇒ x − 2y = 10.

Step 3 — Subtract: −y = −20 ⇒ y = 20; then x = 10 + 2(20) = 50.

✦ Answer: Nuri 50 years, Sonu 20 years.

Practice — Pair of Linear Equations in Two Variables

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