Graphical Method of Finding Solution of a Pair of Linear Equations

Q.

Draw the graphs of the equations $x-y+1=0$ and $3x+2y-12=0$. Determine the coordinates of the vertices of the triangle formed by these lines and the $x$-axis and shade the triangular region.

Q.

Question 1 (i)
Form the pair of linear equations in the following problems, and find their solutions graphically.
10 students of Class X took part in a Mathematics quiz. If the number of girls is 4 more than the number of boys, find the number of boys and girls who took part in the quiz.

Q. Solve the following systems of equations:

3x − 7y + 10 = 0
y − 2x − 3 = 0

Q.

Draw the graphs of the equations $x=3,x=5$ and $2x-y-4=0$. Also find the area of the quadrilateral formed by the lines and the $x$-axis.

Q. Question 10
Find the solution of the pair of equations $\frac{x}{10}+\frac{y}{5}-1=0and\frac{x}{8}+\frac{y}{6}=15find\lambda ,ify=\lambda x+5$

Q. Draw the graph of the following pair of linear equations :
$x+3y=6$ and $2x-3y=2$
Find the ratio of the areas of the two triangles formed by first line, $x=0,y=0$ and second line, $x=0,y=0$.

Q. Question 4
The cost of 4 pens and 4 pencils boxes is Rs.100. there times the cost of a pen is Rs. 15 more than the cost of a pencil box, . Form the pair of linear equations for the above situation. Find the cost pen and a pencil box be Rs. y.

Q.

If the cost of $2$ pencils and $3$ erasers is $Rs.9$ and the cost of $4$ pencils and $6$ erasers is $Rs.18$ . Find the cost of each pencil and each eraser.

Q.

If the length and breadth of the room are increased by 1 m each, its area would be increased by $31{\text{m}}^2$. If the length is increased by 1 m and breadth is decreased by 1 m, the area would decrease by $9{\text{m}}^2$ . Find the area of the floor of the room in ${\text{m}}^2$.

1. $200$

2. $209$

3. $250$

4. $150$

Q. At what point will the line x-y = 8 intersect y axis?

Q.

Find the value of ‘a" for which the system of equations 3x + 2y - 4 = 0 and ax - y - 3 = 0, will represent intersecting lines.

1. a = 3/2

2. a = 2/3

3. a ≠ -3/2

4. a ≠ -2/3

Q.

The coordinates of the point in which the line joining the points $(3,5,-7)$ and $(-2,1,8)$is intersected by the plane $yz$ are given by

1. $\left(0,\frac{13}{5},2\right)$

2. $\left(0,\frac{-13}{5},-2\right)$

3. $\left(0,\frac{-13}{5},\frac{2}{5}\right)$

4. $\left(0,\frac{13}{5},\frac{2}{5}\right)$

Q. Question 4 (i)
Form the pair of linear equations in the following problems and find their solutions (if they exist) by any algebraic method:
A part of monthly hostel charges is fixed and the remaining depends on the number of days one has taken food in the mess. When a student A takes food for 20 days she has to pay Rs 1000 as hostel charges whereas a student B, who takes food for 26 days, pays Rs 1180 as hostel charges. Find the fixed charges and the cost of food per day.

Q. Question 4
The pair of equation y = 0 and y = - 7 has
(A) one solution
(B) two solutions
(C) infinitely many solutions
(D) no solution​​​​​​​

Q.

The graphs of the equations 2x+3y-2=0 and x-2y-8=0 are two lines which are

(a) coincident

(b) parallel

(c) intersecting exactly at one point

(d) perpendicular to each other

Q. Show graphically that each one of the following systems of equations has infinitely many solutions:

x − 2y + 11 = 0
3x − 6y + 33 = 0

Q. What is the solution for the two lines given in the graph?

1. $x=0,y=0$
2. $x=4,y=2$
3. $x=5,y=2$
4. $x=-5,y=2$

Q. Solve the following systems of equations:

$$\begin{gathered} \frac{2}{x}+\frac{5}{y}=1 \\ \frac{60}{x}+\frac{40}{y}=19,x\ne 0,y\ne 0 \end{gathered}$$

Q.

Solve each of the following systems of equations graphically:

$3x+2y=4,2x-3y=7.$

Q.

Show graphically that each of the following given systems of equations is inconsistent.i.e., has no solution:

$2x+3y=4,4x+6y=12.$

Q.

Solve each of the following systems of equations graphically:

$2x-3y+13=0,3x-2y+12=0.$

Q. Draw the graph of $y=2x^2$ and hence solve $2x^2+x-6=0$ .

1. {2, 1.5}
2. {-2, 1.5}
3. {-2, -1.5}
4. {2, -1.5}

Q. If $\frac{5+2\sqrt{3}}{7+4\sqrt{3}}=$$a+b\sqrt{3}$ , then find the value of a and b

Q.

Find the values of a and b for which the following system of equations has infinitely many solutions:

(i) (2a - 1) x - 3y = 5

3x + (b - 2) y = 3

(ii) 2x - (2a + 5) y = 5

(2b + 1) x - 9y = 15

(iii) (a - 1) x + 3y = 2

6x + (1 - 2b) y = 6

(iv) 3x + 4y = 12

(a + b) x + 2 (a - b) y = 5a - 1

(v) 2x + 3y = 7

(a - b) x + (a + b) y = 3a + b - 2

(vi) 2x + 3y - 7 = 0

(a - 1) x + (a + 1) y = (3a - 1)

(vii) 2x + 3y = 7

(a - 1) x + (a + 2) y = 3a

(viii) x + 2y = 1

(a - b) x + (a + b) y = a + b - 2

(ix) 2x + 3y = 7

2ax + ay = 28 - by

Q. Solve the following pair of equations by reducing them to a pair of linear equations:
$\frac{1}{x}-\frac{4}{y}=2\frac{1}{x}-\frac{3}{y}=9$

Q.

Question 7
How many solution(s) of the equation 2x + 1 = x - 3 are there on the:
(i) number line?
(ii) Cartesian plane?

Q. Solve the following systems of equations:

152x − 378y = −74
−378x + 152y = −604

Q.

The number of solutions of equations represented by this graph is

1. No Solution

2. Infinite

3. 2

4. 1

Q. Question 7
Draw the graphs of the equations x - y + 1 = 0 and 3x + 2y - 12 = 0. Determine the coordinates of the vertices of the triangle formed by these lines and the x-axis, and shade the triangular region.

Q. Question 3
The cost of 2 kg of apples and 1kg of grapes on a day was found to be Rs 160. After a month, the cost of 4 kg of apples and 2 kg of grapes is Rs 300. Represent the situation algebraically and geometrically