Representing a Pair of Linear Equations in Two Variables Graphically

Q.

On comparing the ratios$\frac{a_1}{a_2},\frac{b_1}{b_2},\frac{c_1}{c_2}$, find out whether the lines representing the following pairs of linear equations intersect at a point, or parallel or coincident:

$\begin{array}{rcl}9x+3y+12& =& 0\\ 18x+6y+24& =& 0\end{array}$

Q.

The system of equations $kx+y+z=1$, $x+ky+z=k$ and $x+y+zk=k^2$ has no solution if $k$ is equal to

1. $-2$

2. $-1$

3. $1$

4. $0$

Q.

If a pair of linear equations is consistent, then the lines will be __________

1. parallel

2. always coincident

3. intersecting or coincident

4. always intersecting

Q.

For the system of linear equations: $x-2y=1,x-y+kz=-2,ky+4z=6,k\in R$

Consider the following statements:

(A) The system has a unique solution if $k\ne 2,k\ne -2.$

(B) The system has a unique solution if$k=-2$

(C) The system has a unique solution if $k=2$

(D) The system has no solution if $k=2.$

(E) The system has an infinite number of solutions if $k\ne -2.$

Which of the following statements are correct?

1. (B) and (E) only

2. (C) and (D) only

3. (A) and (D) only

4. (A) and (E) only

Q.

For which values of $a$ and $b$, will the following pair of linear equations have infinitely many solutions? $x+2y=1$ and $(a-b)x+(a+b)y=a+b-2$

Q. Figure 2.23 show the graph of the polynomial f(x) = ax² + bx + c for which

(a) a < 0, b > 0 and c > 0
(b) a < 0, b < 0 and c > 0
(c) a < 0, b < 0 and c < 0
(d) a > 0, b > 0 and c < 0

Q. Consider two equations in the variables x and y written in the standard form:
$5x+6y+4=0and$
$10x+12y+7=0$
What is the nature of these two lines?

1. Coincident
2. Intersecting
3. Parallel
4. Coincident or parallel

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. Find graphically the coordinates of the vertices of triangle, whose sides have the equations $y=x-3,2y=x-4andx-4=0.$

1. 
   $(2,-1),(4,1),(4,0)$
2. 
   $(3,-1),(4,1),(4,0)$
3. 
   $(2,-1),(5,1),(4,0)$
4. 
   $(2,-1),(4,1),(0,0)$

Q. If both 'x' and 'y' are natural numbers, the equation 2x + 5y = 7 will have a unique solution.

1. True
2. False

Q.

The pair of equations $y=0$ and $y=-7$ has,

1. One solution

2. Two solutions

3. Infinitely many solutions

4. No solution

Q.

If in general quadratic equation $f\left(x,y\right)=0,∆=0,$ and $a+b=0$, then the equation represents

1. Two parallel straight lines

2. Two perpendicular straight lines

3. Two lines passing through the origin

4. None of these

Q.

Find the image of point P (–3, 1) in the line 2 – 3 = 0.

1. (-3, 0)

2. (-3, 2)

3. (4, -1)

4. (2, 0)

Q.

Use graph paper for the question.

The points A(2, 3), B(4 , 5) & C(7, 2) are the vertices of ∆ ABC.

Write the coordinates of A’B’C’, if ∆ A’B’C’ is the image of ∆ABC, when reflected in the origin.

1. A’(-2, -3), B’(-7, -2), C’(4, -5)

2. A’(-2, -3), B’(-7, 2), C’(4, -5)

3. A’(-2, -3), B’(-7, 2), C’(-4, -5)

4. A’(-2, -3), B’(-7, -2), C’(-4, -5)

5. A’(-2, 3), B’(-7, -2), C’(-4, -5)

Q.

For a pair of linear equation $a_1$=12, $b_1$= 2, $c_1$= 6 and $a_2$=24, $b_2$=4, $c_2$=7 ; the linear equation has

1. Four solutions

2. Infinite solutions

3. Two solutions

4. No solution

Q. On comparing the ratios $\frac{a_1}{a_2},\frac{b_1}{b_2}$ and $\frac{c_1}{c_2}$, find out whether the lines representing the following pairs of linear equations intersect at a point, or are parallel or coincident.
$9x+3y+12=0;18x+6y+24=0$

1. Intersect at a point
2. Parallel
3. Coincident
4. Cannot be determined

Q. You have been asked to plot graphs of the following equations:
Equation 1: $3x+2y=1$
Equation 2: $y-2x=2$
Fill the following tables (find p, q, r, s) which you will use to plot the graphs:
Equation 1:
$\begin{array}{ccc}x& 0& 5\\ y& p& q\end{array}$
Equation 2:
$\begin{array}{ccc}x& r& s\\ y& 0& 5\end{array}$

1. $p=\frac{1}{2},q=-7,r=-1,s=\frac{3}{2}$
2. $p=2,q=7,r=2,s=3$
3. $p=3,q=\frac{1}{2},r=\frac{2}{3},s=0$
4. $p=3,q=5,r=2,s=8$

Q. Question 2 (i)
On comparing the ratios $\frac{a_1}{a_2}$ , $\frac{b_1}{b_2}$ and $\frac{c_1}{c_2}$ , find out whether the lines representing the following pairs of linear equations intersect at a point, are parallel or coincident.
5x - 4y + 8 = 0
7x + 6y - 9 = 0

Q. For what value of $k$, do the equations $3x-y+8=0$ and $6x-ky=-16$ represent coincident lines?

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

Q.

For what value of $k$ , will the system of equations $x+2y=5,3x+ky+15=0$has

a) unique solution

b) no solution

Q.

Determine the line containing the point (2, 3)

1. $2x+3y=5$

2. $3x-2y=0$

3. $2x+3y=12$

4. $4x-3y=1$

Q.

The linear equation $3x+1=2$ has one solution.

1. True
2. False

Q.

Determine the values of $k$ so that the following linear equations have no solutions

$(3k+1)x+3y-2=0;(k^2+1)x+(k-2)y-5=0$

Q.

Whats the point that is on a x = 5 and has ordinate as -3

Q.

Solve the following systems of equations:

$x+2y=\frac{3}{2}$

$2x+y=\frac{3}{2}$

Q.

Question 3 (iii)
Are the following pair of linear equations consistent? Justify your answer
2a + by = a and 4a + 2by -2a = 0; a, b $\ne$ 0

Q. Which point in the graph shows (3, -14)?

1. A
2. B
3. C
4. D

Q. For a pair of linear equations:
$a_{1}x+b_{1}y=c_1$ and $a_{2}x+b_{2}y=c_2$

If $\frac{a_1}{a_2}$ = $\frac{b_1}{b_2}$ = $\frac{c_1}{c_2}$, then the pair of equations have ______ solution(s).

1. zero
2. infinite
3. one
4. two

Q. Question 3 (iii)
Are the following pair of linear equations consistent? Justify your answer
2a + by = a and 4a + 2by -2a = 0; a, b $\ne$ 0

Q. Draw the graphs of the equations $5x-y=5$ and $3x-y=3$. Determine the co-ordinates of the vertices of the triangle formed by these lines and the y axis.