If \(3x-ky+8 = 0\) and \(9x-18y+20 = 0\) have no solution, find k.
6
The two lines are \(3x-ky+8=0\) and \(9x-18y+20=0\), and they must have no solution.
A pair of linear equations \(a_{1}x+b_{1}y+c_{1}=0\) and \(a_{2}x+b_{2}y+c_{2}=0\) has no solution (parallel, non-coincident lines) exactly when \(\frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}}\neq \frac{c_{1}}{c_{2}}\).
Match coefficients: \(a_{1}=3,\;b_{1}=-k,\;a_{2}=9,\;b_{2}=-18\).
Set the first two ratios equal: \(\frac{3}{9}=\frac{-k}{-18}\), i.e. \(\frac{1}{3}=\frac{k}{18}\).
Cross-multiplying, \(k=\frac{18}{3}=6\).
Verify the constant ratio stays different: \(\frac{c_{1}}{c_{2}}=\frac{8}{20}=\frac{2}{5}\neq \frac{1}{3}\), confirming the lines are parallel and never meet.
The condition for no solution is equal coefficient ratios but a different constant ratio. Hence the value of k is 6.
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If 3x - ky + 8 = 0 and 9x - 18y + 20 = 0 have no solution, find k.
Important Questions from Linear Equation in 2 or more Variables
Six years ago, the ratio of ages of A to B was 7 ∶ 5. After 4 years from now, the ratio of their ages will be 11 ∶ 9. What is A’s age at present?
7p - [3q - {8p - (4q - 10p)}] = ?
Surya is 25 years older than his son. In 5 years, he will be twice as old as his son. What will be Surya’s age after 3 years?
The difference between the ages of two sisters is 2 years when father’s age is 52. Father is elder by 2 years to mother. Elder sister’s age is half of mother’s age. Find the age of younger sister?
The sum of the digits of a 2 digit number is 9, When 27 is added to the number, the digits get interchanged. Find the number.
A. 45
B. 36
C. 18
D. 27
