Quantitative Reasoning

Linear Equation

Medium

The value of k, for which the system of equations 3x-ky-20 = 0, and 6x-10y+40 = 0 has no solution, is:

✅ Correct Option: 3

For two linear equations to have no solution, the lines must be parallel.

For equations:

$a_1x + b_1y + c_1 = 0$
$a_2x + b_2y + c_2 = 0$

The condition for no solution is:

$\dfrac{a_1}{a_2} = \dfrac{b_1}{b_2} \neq \dfrac{c_1}{c_2}$

From $3x - ky - 20 = 0$ :

$a_1 = 3$

$b_1 = -k$

$c_1 = -20$

From $6x - 10y + 40 = 0$ :

$a_2 = 6$

$b_2 = -10$

$c_2 = 40$

Calculate $\dfrac{a_1}{a_2}$ :

$\dfrac{a_1}{a_2} = \dfrac{3}{6} = \dfrac{1}{2}$

For no solution, $\dfrac{b_1}{b_2} = \dfrac{a_1}{a_2}$ :

$\dfrac{b_1}{b_2} = \dfrac{-k}{-10} = \dfrac{k}{10}$

Setting equal to $\dfrac{a_1}{a_2}$ :

$\dfrac{k}{10} = \dfrac{1}{2}$

$k = 10 \times \dfrac{1}{2}$

$k = 5$

Check $\dfrac{c_1}{c_2}$ :

$\dfrac{c_1}{c_2} = \dfrac{-20}{40} = -\dfrac{1}{2}$

Since $\dfrac{1}{2} = \dfrac{1}{2} \neq -\dfrac{1}{2}$ , the condition for no solution is satisfied.

Therefore, $k = 5$

2025: 14 May Shift 2

The graphs of ax+by = c and dx+ey = f will be:

(A) parallel, if the system has no solution.

(B) coincident, if the system has finite numbers of solutions.

Which of the above statements is/ are correct?

Choose the correct answer from the options given below:

2024

The present age of Harish is 8 times the sum of the ages of his two sons at present. After 8 years, his age will be 2 times the sum of the ages of his two sons. The present age of Harish (in years) is:

2025: 27 May Shift 2

The pair of linear equations mx + 2y + 3 = 0 and 3x + 6y + 2 = 0 intersect each other, if

2025: 30 May Shift 2

For what value of k, the system of equations $kx + 4y - k + 4 = 0$ and $16x + ky = k$ has an infinite number of solutions?

2025: 29 May Shift 1

The present ratio of ages of P and Q is 2:7. Six years ago, this ratio was 1:5. Determine the present age of P?

2025: 19 May Shift 1

The sum of digits of a two-digit number is 14 and the difference between the two digits of the number is 2. The product of the two digits of the number is:

The value of k, for which the system of equations 3x-ky-20 = 0, and 6x-10y+40 = 0 has no solution, is:

Solution

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