Conditions for a system of linear equations to have infinite solutions

Q. For a system of Linear Equations to have infinite solutions , the ratio of the coefficients in standard form of ax + by + c= 0 must be

1. $\frac{a1}{a2}=\frac{b1}{b2}\ne \frac{c1}{c2}$
2. $\frac{a1}{a2}\ne \frac{b1}{b2}\ne \frac{c1}{c2}$
3. $\frac{a1}{a2}\ne \frac{b1}{b2}=\frac{c1}{c2}$
4. $\frac{a1}{a2}=\frac{b1}{b2}=\frac{c1}{c2}$

Q. Let $S$ be the set of all $\lambda$ $\in R$ for which the system of linear equations
$2x-y+2z=2$
$x-2y+\lambda z=-4$
$x+\lambda y+z=4$
has no solution. Then the set $S$

1. is an empty set.
2. is a singleton.
3. contains more than two elements.
4. contains exactly two elements.

Q. $4-3y=6y+4-9y$
Which of the following best describes the solution set the equation shown above?

1. The equation has no solution
2. The equation has infinitely many solution
3. The equation has exactly one solution, $y=0$
4. The equation has exactly one solution, $y=\frac{4}{3}$

Q. Examine the consistency of the system of equations$5x-y+4z=5,2x+3y+5z=2,5x-2y+6z=-1$

Q. Solve the following system of equations.

1. $a-b,-2ab$
2. $a+b,-2ab$
3. $a^2+b^2,\frac{-2ab}{a+b}$
4. $a+b,\frac{-2ab}{a+b}$

Q. For the equations $x+2y+3z=1,2x+y+3z=2$ and $5x+5y+9z=4$

1. There exists infinitely many solution
2. There is only one solution
3. None of the above
4. There is no solution

Q. 3x − 2y = a ;
−15x + 10y = −7
In the system of equations above, a and b are constants. If the system has infinitely many solutions, what is the value of a?

1. $\frac{7}{5}$
2. $\frac{-5}{7}$
3. $\frac{4}{7}$
4. $\frac{-7}{4}$

Q. The equation $2x+5y=7$ has a unique solution, if $x,y$ are :

1. Natural numbers
2. Positive real numbers
3. Real numbers
4. None of these

Q. Find the solutions for the following pairs of equation and identify the common solution
$3x+2y=6$ and $5x-2y=10$

Q. $-x=-7y-8$ , $x-7y=k$
Consider the system of equations above. Which of the following choices of $k$ will result in a system of equations will infinitely many solutions?

1. Any number
2. Any number except 8
3. 8
4. $-8$