Solutions: Linear Equations - Test 4

Q1. (b) 2

Multiply first eq by 3 and second by 2: $6x+9y=36$ and $6x-4y=10$. Subtract: $13y=26 \Rightarrow y=2$.

Q2. (c) 3

Adding both equations: $60x + 60y = 180 \Rightarrow 60(x+y) = 180 \Rightarrow x+y=3$.

Q3. (a) intersecting at (a, b)

$x=a$ is vertical line, $y=b$ is horizontal line. They meet at $(a, b)$.

Q4. (a) Both A and R are true and R is the correct explanation of A.

$\frac{1}{2} = \frac{2}{4} \neq \frac{3}{7}$. Parallel lines, hence inconsistent.

Q5. $3x - 5y = 4$ ...(i), $9x - 2y = 7$ ...(ii)
Multiply (i) by 3: $9x - 15y = 12$. Subtract (ii): $-13y = 5 \Rightarrow y = -5/13$.
Substitute in (i): $3x - 5(-5/13) = 4 \Rightarrow 3x + 25/13 = 4 \Rightarrow 3x = 27/13 \Rightarrow x = 9/13$.
x = 9/13, y = -5/13 [2]

Q6. $s = t + 3$. Substitute in second eq: $\frac{t+3}{3} + \frac{t}{2} = 6$.
Multiply by 6: $2(t+3) + 3t = 36 \Rightarrow 2t + 6 + 3t = 36 \Rightarrow 5t = 30 \Rightarrow t = 6$.
$s = 6 + 3 = 9$. s = 9, t = 6 [2]

Q7. Add eqs: $200(x+y) = 1000 \Rightarrow x+y=5$.
Subtract eqs: $-2x + 2y = -2 \Rightarrow -x+y=-1$.
Solving $x+y=5$ and $-x+y=-1$: $2y=4 \Rightarrow y=2, x=3$.
x = 3, y = 2 [2]

Q8. Let $1/\sqrt{x} = u, 1/\sqrt{y} = v$.
$2u + 3v = 2$ and $4u - 9v = -1$.
Multiply first by 3: $6u + 9v = 6$. Add to second: $10u = 5 \Rightarrow u = 1/2$.
$2(1/2) + 3v = 2 \Rightarrow 1 + 3v = 2 \Rightarrow 3v = 1 \Rightarrow v = 1/3$.
$\sqrt{x} = 2 \Rightarrow x = 4$. $\sqrt{y} = 3 \Rightarrow y = 9$. [3]

Q9. Multiply first by $b$, second by $a$:
$abx + b^2y = b(a-b)$
$abx - a^2y = a(a+b)$
Subtract: $(b^2+a^2)y = ab - b^2 - a^2 - ab = -(a^2+b^2)$.
$y = -1$. Substitute in first: $ax - b = a - b \Rightarrow ax = a \Rightarrow x = 1$.
x = 1, y = -1 [3]

Q10. Let boat speed = $x$, stream speed = $y$.
$\frac{30}{x-y} + \frac{44}{x+y} = 10$ ...(i)
$\frac{40}{x-y} + \frac{55}{x+y} = 13$ ...(ii)
Let $1/(x-y) = u, 1/(x+y) = v$.
$30u + 44v = 10$ and $40u + 55v = 13$.
Solving gives $u = 1/5, v = 1/11$.
$x-y = 5, x+y = 11$. Solving these: $2x = 16 \Rightarrow x = 8, y = 3$.
Boat: 8 km/h, Stream: 3 km/h [5]

(i) Let unit digit $y$, tens digit $x$. Number $10x+y$.
$x + y = 9$
$9(10x+y) = 2(10y+x) \Rightarrow 90x+9y = 20y+2x \Rightarrow 88x - 11y = 0 \Rightarrow 8x - y = 0$. [2]

(ii) $y = 8x$. Substitute in $x+y=9$: $x+8x=9 \Rightarrow 9x=9 \Rightarrow x=1$.
$y = 8$. Number is 18. [1]

(iii) $18 + 27 = 45$. [1]