Solve the following systems of equations graphically:

Question 1: $x+y=3 \ \ \ \ \ \& \ \ \ \ \ 2x+5y= 12$

To draw the line $x+y=3$

 x 0 3 y 3 0

To draw the line $2x+5y= 12$

 x 0 6 y 2.4 0

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Question 2: $x-2y =5 \ \ \ \ \ \& \ \ \ \ \ 2x+3y =10$

To draw the line $x-2y =5$

 x 0 5 y -2.5 0

To draw the line $2x+3y =10$

 x 0 5 y 3.33 0

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Question 3: $3x+y+1=0 \ \ \ \ \ \& \ \ \ \ \ 2x-3y+8=0$

To draw the line $3x+y+1=0$

 x 0 -0.33 y -1 0

To draw the line $2x-3y+8=0$

 x 0 -4 y 8/3 0

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Question 4: $2x+y-3=0 \ \ \ \ \ \& \ \ \ \ \ 2x-3y-7 =0$

To draw the line $2x+y-3=0$

 x 0 1.5 y 3 0

To draw the line $2x-3y-7 =0$

 x 0 3.5 y -(7/3) 0

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Question 5: $x+y = 6 \ \ \ \ \ \& \ \ \ \ \ x-y=2$

To draw the line $x+y = 6$

 x 0 6 y 6 0

To draw the line $x-y=2$

 x 0 2 y -2 0

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Question 6: $x-2y =6 \ \ \ \ \ \& \ \ \ \ \ x-6y =3$

To draw the line $x-2y =6$

 x 0 6 y -3 0

To draw the line $3x-6y =0$

 x 0 3 y -0.5 0

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Question 7: $x+y=4 \ \ \ \ \ \& \ \ \ \ \ 2x-3y=3$

To draw the line $x+y=4$

 x 0 4 y 4 0

To draw the line $2x-3y=3$

 x 0 1.5 y -1 0

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Question 8: $2x+3y=4 \ \ \ \ \ \& \ \ \ \ \ x -y + 3=0$

To draw the line $2x+3y=4$

 x 0 2 y 1.33 0

To draw the line $x -y + 3=0$

 x 0 -3 y 3 0

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Question 9: $2x-3y+13=0 \ \ \ \ \ \& \ \ \ \ \ 3x-2y+12=0$

To draw the line $2x-3y+13=0$

 x 0 -6.5 y 4.33 0

To draw the line $3x-2y+12=0$

 x 0 -4 y 6 0

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Question 10: $2x+3y+5=0 \ \ \ \ \ \& \ \ \ \ \ 3x-2y-12=0$

To draw the line $2x+3y+5=0$

 x 0 -2.5 y -(5/3) 0

To draw the line $3x-2y-12=0$

 x 0 4 y -6 0

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

Question 11: $2x+3y =6 \ \ \ \ \ \& \ \ \ \ \ 4x+6y =12$

To draw the line $2x+3y =6$

 x 0 3 y 2 0

To draw the line $4x+6y =12$

 x 0 3 y 2 0

These are the same line. They over lap each other. Hence there are infinite solutions.

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Question 12: $3x+y=8 \ \ \ \ \ \& \ \ \ \ \ 6x+2y=16$

To draw the line $3x+y=8$

 x 0 (8/3) y 8 0

To draw the line $6x+2y=16$

 x 0 (8/3) y 8 0

These are the same line. They over lap each other. Hence there are infinite solutions.

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Question 13: $x-2y =5 \ \ \ \ \ \& \ \ \ \ \ 3x-6y =15$

To draw the line $x-2y =5$

 x 0 5 y -2.5 0

To draw the line $3x-6y =15$

 x 0 5 y -2.5 0

These are the same line. They over lap each other. Hence there are infinite solutions.

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Question 14: $x-2y+11=0 \ \ \ \ \ \& \ \ \ \ \ 3x - 6y + 33 = 0$

To draw the line $x-2y+11=0$

 x 0 -11 y 5.5 0

To draw the line $3x - 6y + 33 = 0$

 x 0 -11 y 5.5 0

These are the same line. They over lap each other. Hence there are infinite solutions.

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Show graphically that each one of the following systems of equations is in-consistent ( i.e. has No solution):

Question 15: $3x-5y =20 \ \ \ \ \ \& \ \ \ \ \ 6x -10y = -40$

To draw the line $3x-5y =20$

 x 0 6.33 y -4 0

To draw the line $6x -10y = -40$

 x 0 -6.33 y 4 0

These are parallel lines. Hence there is no intersection and hence no solution.

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Question 16: $2y-x=9 \ \ \ \ \ \& \ \ \ \ \ 6y -3x =21$

To draw the line $2y-x=9$

 x 0 -9 y 4.5 0

To draw the line $6y -3x =21$

 x 0 7 y 3.5 0

These are parallel lines. Hence there is no intersection and hence no solution.

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Question 17: $x-2y=6 \ \ \ \ \ \& \ \ \ \ \ 3x-6y =0$

To draw the line $x-2y=6$

 x 0 6 y -3 0

To draw the line $3x-6y =0$

 x 0 1 y 0 0.5

These are parallel lines. Hence there is no intersection and hence no solution.

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Question 18: $3x-4y-1=0 \ \ \ \ \ \& \ \ \ \ \ 2x - \frac{8}{3} y + 5 = 0$

To draw the line $3x-4y-1=0$

 x 0 1 y -0.25 0.5

To draw the line $2x - \frac{8}{3} y + 5=0$

 x 0 -2.5 y 1.875 0

These are parallel lines. Hence there is no intersection and hence no solution.

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Question 19: Determine graphically the vertices of the triangle, the equations of whose sides are given below:

(i) $2y - x =8, \ \ \ \ \ 5y - x =14 \ \ \ \ \ \& \ \ \ \ \ y -2x =1$

(ii) $y = x, \ \ \ \ \ y=0 \ \ \ \ \ \& \ \ \ \ \ 3x + 3y = 10$

i)  Triangle ABC.

ii) Triangle ABC

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Question 20: Determine, graphically whether the system of equations $x - 2y = 2, \ \ \ \ \ 4x - 2y = 5$ is consistent or in-consistent.

Consistent equations. Point of intersection is (1, -0.5)

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Question 21: 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) $2y=4x-6 \ \ \ \ \ \& \ \ \ \ \ 2x-y+3=0$

i) The point of intersection is (1.8, -0.8).  Hence there is a unique solution.

ii)  The lines are parallel. Hence no solution.$\\$

Question 22: Solve graphically each of the following systems of linear equations. Also find the coordinates of the points where the lines meet axis of y.

1. $2x-5y+4=0 \ \ \ \ \ \& \ \ \ \ \ 2x+y-8=0$
2. $3x + 2y = 12 \ \ \ \ \ \& \ \ \ \ \ 5x-2y=4$
3. $2x+y-11=0 \ \ \ \ \ \& \ \ \ \ \ x - y - 1 = 0$
4. $3x + y - 5 = 0 \ \ \ \ \ \& \ \ \ \ \ 2x-y-5=0$
5. $x+2y-7 =0 \ \ \ \ \ \& \ \ \ \ \ 2x-y-4=0$
6. $2x-y-5=0 \ \ \ \ \ \& \ \ \ \ \ x -y - 3 = 0$

Question 23: Solve the following system of linear equations graphically and shade the region between the two lines and x-axis:

i) $2x + 3y = 12 \ \ \ \ \ \& \ \ \ \ \ x-y=1$

ii) $3x+2y-11 =0 \ \ \ \ \ \& \ \ \ \ \ 2x-3y+10=0$

iii) $3x + 2y -4=0 \ \ \ \ \ \& \ \ \ \ \ 2x-3y-7=0$

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Question 24: the graphs of the following equations on the same graph paper: $2x+3y =12, \ \& \ x-y=1$. Find the coordinates of the vertices of the triangle formed by the two straight lines and the y-axis ($x=0$).

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Question 25: Draw the graphs of $x-y+1=0$ and $3x+2y-12=0$. Determine the coordinates of the vertices of the triangle formed by these lines and x- axis and shade the triangular area. Calculate the area bounded by these lines and x-axis.

Area of the triangle $=$ $\frac{1}{2}$ $\times 3 \times 3 +$ $\frac{1}{2}$ $\times 3 \times 2 = 4.5 + 3 = 7.5$ sq. units.

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Question 26: Solve graphically the system of linear equations: $4x-3y+4=0 \ \ \ \ \ \& \ \ \ \ \ 4x+3y-20=0$ Find the area bounded by these lines and x-axis.

Area of the triangle $=$ $\frac{1}{2}$ $\times 4 \times 3 +$ $\frac{1}{2}$ $\times 4 \times 3 = 6 + 6 = 12$ sq. units.

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Question 27: Solve the following system of linear equations graphically: $3x+y-11=0 \ \ \ \ \ \& \ \ \ \ \ x-y-1=0$ shade the region bounded by these lines and y -axis. Also, find the area of the region bounded by the these lines and y-axis.

Area of the triangle $=$ $\frac{1}{2}$ $\times 2 \times 12 = 12$ sq. units.

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Question 28: Solve graphically each of the following systems of linear equations. Also, find the coordinates of the points where the lines meet the axis of $x$ in each system:

1. $2x+y=6 \ \ \ \ \ \& \ \ \ \ \ x-2y= -2$
2. $x+2y=5 \ \ \ \ \ \& \ \ \ \ \ 2x-3y=-4$
3. $2x-y=2 \ \ \ \ \ \& \ \ \ \ \ 4x-y=8$
4. $2x + 3y = 8 \ \ \ \ \ \& \ \ \ \ \ x -2y = -3$

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Question 29: Draw the graphs of the following equations: $2x-3y +6 = 0 \ \ \ \ \ 2x+3y-18=0 \ \ \ \ \ \& \ \ \ \ \ y-2=0$  Find the vertices of the triangle so obtained. Also, find the area of the triangle.

Area of the triangle $=$ $\frac{1}{2}$ $\times 6 \times 2 = 6$ sq. units.

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Question 30: Solve the following system of equations graphically:  $2x-3y+6=0 \ \ \ \ \ \& \ \ \ \ \ 2x+3y-18=0$ Also, find the area of the region bounded by these two lines and y-axis.

Area of the triangle $=$ $\frac{1}{2}$ $\times 4 \times 3 = 6$ sq. units.

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Question 31: Solve the following system of linear equations graphically: $4x-5y -20 =0 \ \ \ \ \ \& \ \ \ \ \ 3x+5y-15=0$ Determine the vertices of the triangle formed by the lines representing the above equation and the y-axis.

Area of the triangle $=$ $\frac{1}{2}$ $\times 7 \times 5 = 17.5$ sq. units.

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Question 32: Draw the graphs of the equations $5x -y =5$ and $3x - y =3$ Determine the coordinates of the vertices of the triangle formed by these and y-axis. Calculate the area of the triangle so formed.

Area of the triangle $=$ $\frac{1}{2}$ $\times 2 \times 1 = 1$ sq. units.

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Question 33: Form the pair of linear equations in the following problems, and find their solution graphically:

(i) 10 students of class X took part in 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.

(ii) 5 pencils and 7 pens together cost Rs. 50, whereas 7 pencils and 5 pens together cost Rs. 46. Find the cost of one pencil and a pen.

(iii) Champa went to a sale to purchase some pants and skirts. When her friends asked her how many of each she had bought, she answered, “The number of skirts is two more than twice the number of pants purchased. Also, the number of skirts is four less than four times the number of pants purchased. ” Help her friends to find how many pants and skirts Champa bought.

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Question 34: Solve the following system of equations graphically: Shade the region between the lines and the y-axis

1. $3x-4y =7 \ \ \ \ \ \& \ \ \ \ \ 5x+2y=3$
2. $4x-y=4 \ \ \ \ \ \& \ \ \ \ \ 3x+2y =14$

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Question 35: Represent the following pair of equations graphically and write the coordinates of points where the lines intersects y-axis  $x+3y=6 \ \ \ \ \ \& \ \ \ \ \ 2x -3y =12$

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Question 36: 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 (ii) Parallel lines (iii) coincident lines

i) Intersecting lines: The slope should be different. $3y + 3x = 8$

ii) Parallel lines: Slope should be equal. $3y + 2x = 16$

iii) Coincident lines: Should be overlapping. $6y + 4x = 16$

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Question 37: Determine graphically the coordinates of the vertices of a triangle, the equations of whose sides are:
(i) $y = x, \ \ \ \ \ y =2x \ \ \ \ \& \ \ \ \ \ y+ x = 6$
(ii) $y=x, \ \ \ \ \ 3y=x \ \ \ \ \ \& \ \ \ \ \ x+y=8$

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Question 38: Graphically, solve the following pair of equations: $2x + y = 6 \ \ \ \ \ \& \ \ \ \ \ 2x-y + 2 = 0$Find the ratio of the areas of the two triangles formed by the lines representing these equations with the x-axis and the lines with the y-axis.

Ratio of Areas $=$ $\frac{\frac{1}{2} \times 4 \times 1}{\frac{1}{2} \times 4 \times 4}$ $=$ $\frac{1}{4}$

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Question 39: Determine, graphically, the vertices of the triangle formed by the lines $y=x, \ \ \ \ \ 3y=x \ \ \ \ \ \& \ \ \ \ \ x+y=8$.

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Question 40: Draw the graph of the equations $x = 3, \ \ \ \ \ x = 5 \ \ \ \ \ \& \ \ \ \ \ 2x - y - 4 = 0$. Also, find the area of the quadrilateral formed by the lines and the x-axis.

Area $= 2 \times 2 +$ $\frac{1}{2}$ $\times 2 \times 4 = 4 + 4 = 8$ sq. units

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Question 41: Draw the graphs of the lines $x = -2$, and $y = 3$. Write the vertices of the figure formed by these lines, the x-axis and the y-axis. Also, find the area of the figure.

Area $= 2 \times 3 = 6$ sq. units
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Question 42: Draw the graphs of the pair of linear equations $x-y+2=0$ and $4x - y - 4 = 0$. Calculate the area of the triangle formed by the lines so drawn and the x-axis.
Area of the triangle $=$ $\frac{1}{2}$ $\times 3 \times 4 = 6$ sq. units.
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