Comparing the Ratios of Coefficients of a Linear Equation

Q. Find k for which the system x + 2y = 3 and 5x + ky + 7 = 0 is inconsistent.

Q. For what value of k will the following system of linear equations has no solution:

3x + y = 1

(2k − 1)x + (k − 1)y = 2k + 1

Q.

The system kx-y=2 and 6x-2y=3 has a unique solution only when

(a) k=0 (b)$k\ne 0$ (c) k=3 (d) $k\ne 3$

Q.

Question 8
The value of c for which the pair of equations cx – y = 2 and 6x – 2y = 3 will have infinitely many solutions is
(A) 3
(B) – 3
(C) –12
(D) no value

Q.

Which of the following pairs of linear equations are consistent/inconsistent?

If consistent, obtain the solution graphically:

$x+y=5,2x+2y=10$.

Q. Find the value of a and b .
2x+3y=7 and 2ax+ay=28-by

Q.

For what value of k does the system of equations $x+2y=5,3x+ky+15=0$ have (i) a unique solution, (ii) no solution?

Q.

Find the value of k for which each of the following systems of linear equations has an infinite number of solutions:

$2x+3y=7,(k-1)x+(k+2)y=3k.$

Q. If the system of equations kx − 5y = 2, 6x + 2y = 7 has no solution, then k =

(a) −10
(b) −5
(c) −6
(d) −15

Q.

For which values of p and q, will the following pair of linear equations have infinitely many solutions? 4x + 5y = 2 , (2p + 7q) x + (p + 8q) y = 2q-p + 1

Q.

Question 6 (i)
Given the linear equation 2x + 3y - 8 = 0, write another linear equation in two variables such that the geometrical representation of the pair so formed is:
(i) Intersecting lines

Q.

Find the value $k$ for which each of the following systems of equations has a unique solution:

$4x-5y=k,2x-3y=12.$

Q. Show that there is no value of n for which (2ⁿ × 5ⁿ) ends in 5.

Q. Determine graphically the vertices of the triangle, the equations of whose sides are given below :

(i) 2y − x = 8, 5y − x = 14 and y − 2x = 1
(ii) y = x, y = 0 and 3x + 3y = 10

Q.

The system x-2y=3 and 3x+ky=1 has a unique solution only when

(a) k=-6 (b)$k\ne -6$ (c) k=0 (d) $k\ne 0$

Q. Find the solution of the given pair of equations
$2x+3y=9,3x+4y=5$

Q. Determine, graphically whether the system of equations x − 2y = 2, 4x − 2y = 5 is consistent or in-consistent.

Q.

Write the value of $k$ for which the system of equations $3x-2y=0$ and $kx+5y=0$ has infinitely many solutions.

Q. Determine the value of k so that the following linear equations have no solution:

(3k + 1) x + 3y − 2 = 0

(k² + 1) x + (k − 2) y − 5 = 0

Q. Determine, by drawing graphs, whether the following system of linear equations has a unique solution or not :

(i) 2x − 3y = 6, x + y = 1
(ii) 2y = 4x − 6, 2x = y + 3

Q.

Find the value of k for which each of the following systems of equations has no solution:

$8x+5y=9,kx+10y=15.$

Q.

Find value of k for infinite Sol for pair of linear equation

kx + 3y = 2k + 1

2 ( k + 1)x + 9x = 7k + 1

Q.

Find the value for k which each of the following systems of equations has a unique solution:

$2x+3y-5=0,kx-6y-8=0.$

Q.

Find the value of k for which the system 3x+5y=0, kx+10y=0 has a nonzero solution.

Q. Solve the following systems of equations graphically:

x − 2y = 5
2x + 3y = 10

Q.

The following system of linear equations is,

$$\begin{gathered} 2\mathrm{x}+\; 3\mathrm{y}+\; 2\mathrm{z}=\; 9 \\ 3\mathrm{x}+\; 2\mathrm{y}+\; 2\mathrm{z}=\; 9 \\ \mathrm{x}-\mathrm{y}+\; 4\mathrm{z}=\; 8 \end{gathered}$$

1. does not have any solution

2. has a unique solution

3. has a solution $(\alpha ,\beta ,\gamma )$ satisfying $\alpha +{\beta }^2+{\gamma }^3=12$

4. has infinitely many solutions

Q.

For what value of k does the system of equations $kx+2y=5,3x-4y=10$ have

(i) a unique solution,

(ii) no solution?

Q. If the pair of linear equations 4x + (k + 1)y = 15 and 7x + 8y = 16 has no solution, then the value of k is

1. $\frac{13}{2}$
   
2. $\frac{25}{7}$
   
3. $\frac{15}{2}$
   
4. $\frac{39}{7}$
   

Q. Solve the following systems of equations graphically:

x + y = 3
2x + 5y = 12

Q.

For what value of k do the equations kx-2y=3 and 3x+y=5 represent two lines intersecting at a unique point?

(a) k=3 (b) k=-3

(c) k=6 (d) all real values except -6