General Form of a Pair of Linear Equations in Two Variables

Q. If the pair of linear equations $a_{1}x+b_{1}y+c_1=0$ and $a_{2}x+b_{2}y+c_2=0,(a_1,b_1,c_1,a_2,b_{2}and{c}_2$ are all real numbers and $a_1,b_1,a_2,b_2,$ are not equal to zero) represents parallel lines, then which of the following is correct?

1. $\frac{a_1}{a_2}=\frac{b_1}{b_2}\ne \frac{c_1}{c_2}$
   
2. $\frac{a_1}{a_2}\ne \frac{b_1}{b_2}$
   
3. $\frac{a_1}{a_2}=\frac{2b_1}{b_2}=\frac{2c_1}{c_2}$
   
4. $\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}$
   

Q. In the equation $ax+by+c=0,$ the line is parallel to $x$ axis when $a$ = .

1. $2$
2. $0$
3. $1$

Q. The line $ax+by+c=0,$ is parallel to $x$ axis when $a$ = .

1. $2$
2. $0$
3. $1$

Q. What are the values of a, b and c for the equation $y=0.5x+\sqrt{7}$ when written in the standard form: $ax+by+c=0$ ?

1. $0.5,1,-\sqrt{7}$
2. $0.5,-1,\sqrt{7}$
3. $-0.5,1,\sqrt{7}$
4. $0.5,1,\sqrt{7}$

Q. If am ≠ bl, then the system of equations

$$\begin{gathered} ax+by=c \\ lx+my=n \end{gathered}$$

(a) has a unique solution
(b) has no solution
(c) has infinitely many solutions
(d) may or may not have a solution

Q.

Question 1 (ii)
For which value(s) of $\lambda$ does the pair of linear equations $\lambda x+y={\lambda }^{2}andx+\lambda y=1$ have infinitely many solutions?

Q. Convert the following equations in the general form given by:
$a_{1}x+b_{1}y+c_1=0and$
$a_{2}x+b_{2}y+c_2=0$,
and find $a_1,b_{2}and{c}_2$ respectively.
(1) 4x + 9y = 11
(2) 3x + 3y = 0

1. 3, 3, 11
2. 3, 3, -11
3. 4, 3, 0
4. 4, 3, 11

Q.

Question 1 (iii)
For what value(s) of $\lambda$, does the pair of linear equations $\lambda x+y={\lambda }^{2}andx+\lambda y=1$ have a unique solution?

Q. Convert the following equations in the general form given by:
$a_{1}x+b_{1}y+c_1=0and$
$a_{2}x+b_{2}y+c_2=0$,
and find $a_1,b_{2}and{c}_2$ respectively.
$4x+9y=11$
$3x+3y=0$

1. 3, 3, -11
2. 3, 3, 11
3. 4, 3, 0
4. 4, 3, 11

Q.

1. True
2. False

Q.

The value of x which satisfies following pair of linear equations is:
(i) x + 2y = -2
(ii) x - 2y = 12

1. 5

2. 7

3. -5

4. -7

Q. Two customers Shyam and Ekta are visiting a particular shop in the same week (Tuesday to Saturday). Each is equally likely to visit the shop on any day as on another day. What is the probability that both will visit the shop on different days?

OR

Form the pair of linear equations in the following problems, and find their solutions (if they exist) by the elimination method:
The sum of the digits of a two-digit number is 9. Also, nine times this number is twice the number obtained by reversing the order of the digits. Find the number.

[3 Marks]

Q. If $b=$ in the equation $ax+by+c=0$ where $a,b$ and $c$ are constants, then the line represented by the equation is parallel to Y-axis.

1. 1
2. 2
3. \N

Q.

Condition satisfied by the pair of equation $2x+3y=-5$ and $4x+6y-10=0$ is

1. $\frac{a_1}{a_2}$ ≠ $\frac{b_1}{b_2}$ = $\frac{c_1}{c_2}$.

2. $\frac{a_1}{a_2}$ = $\frac{b_1}{b_2}$ = $\frac{c_1}{c_2}$

3. $\frac{a_1}{a_2}$ = $\frac{b_1}{b_2}$ ≠ $\frac{c_1}{c_2}$

4. $\frac{a_1}{a_2}$ ≠ $\frac{b_1}{b_2}$

Q. Given a linear equation $2x+3y=10$. How many pairs of solution ($x,y$) are possible such that $x$ and $y$ $(\ge 0)$ both are integers?

1. $1$
2. $3$
3. $4$
4. $2$

Q. Question 2 (iii)
Form the pair of linear equations in the following problems, and find their solutions (if they exist) by the elimination method:
The sum of the digits of a two-digit number is 9. Also, nine times this number is twice the number obtained by reversing the order of the digits. Find the number.

Q. If a point $(1,2)$ lies on the line represented by the equation $3y=ax+7,$ then the value of $a$ is .

1. $-1$

2. $1$
3. $2$

Q.

Which of the following are Pair of Linear Equation in two variables?

1. 2x + y = 2; 3x - z=2

2. 12x + 4y + 3 = 0; 4x + 4y + 4 = 0

3. x + 2 = 0; a + 2b = 0

4. x + y - 3 = 0; 4a + 2b + c = 24

Q. Convert the following equations in the general form given by:
$a_{1}x+b_{1}y+c_1=0and$
$a_{2}x+b_{2}y+c_2=0$,
and find $a_1,b_{2}and{c}_2$ respectively.
$4x+9y=11$
$3x+3y=0$

1. 3, 3, -11
2. 4, 3, 0
3. 3, 3, 11
4. 4, 3, 11

Q. The graph for the linear equation given below

Q.

Condition satisfied by the pair of equation $2x+3y=-5$ and $4x+6y-10=0$ is

1. ≠ = .

2. = =

3. = ≠

4. ≠

Q. Which of the following is/are an equation?

1. $x=y$
2. $x=0$
3. $y=0$
4. None of the above

Q. Solve for $x$ and $y$:
$0.4x+0.3y=1.7$,
$0.7x-0.2y=0.8$.

Q. If the system of equation, $a^{2}x-ay=1-a$ & $bx+(3-2b)y=3+a$ possesses a unique solution $x=1$, $y=1$ then:

1. $a=1$, $b=1$
2. $a=-1$, $b=1$
3. $a=0$, $b=0$
4. None of the above

Q. If a point $(1,2)$ lies on the line represented by the equation $3y=ax+7,$ then the value of $a$ is .

1. $-1$

2. $1$
3. $2$

Q.

Which of the following are Pair of Linear Equation in two variables?

1. $2x+y=2;3x-z=2$

2. $12x+4y+3=0;4x+4y+4=0$

3. $x+2=0;a+2b=0$

4. $x+y-3=0;4a+2b+c=24$

Q. Divide 80 into two numbers, such that 5 times one number is equal to 3 times the other number.

1. $30and50$
2. $35and52$
3. $40and60$
4. $22and49$

Q. ax+by-c=0 is the generic form of linear equation in two variables.

1. False
2. True

Q. Based on equations reducible to linear equations
Solve for x and y: $9+25xy=53x$ and $27-4xy=x$

1. $x=1,y=4$
2. $x=2,y=7$
3. $x=6,y=2$
4. $x=3,y=2$

Q.

Condition satisfied by the pair of equation $2x+3y=-5$ and $4x+6y-10=0$ is

1. $\frac{a_1}{a_2}$ ≠ $\frac{b_1}{b_2}$ = $\frac{c_1}{c_2}$.

2. $\frac{a_1}{a_2}$ = $\frac{b_1}{b_2}$ = $\frac{c_1}{c_2}$

3. $\frac{a_1}{a_2}$ = $\frac{b_1}{b_2}$ ≠ $\frac{c_1}{c_2}$

4. $\frac{a_1}{a_2}$ ≠ $\frac{b_1}{b_2}$