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We plot the points for both of the equations to find the solution.
(ii)
Let the cost of one pencil=Rs.X
and Let the cost of one pen=Rs.Y
According to the given conditions, we have:
=5x + 7y = 50
=5x=50-7y
=x=10-7/5y
Three solutions of this equation can be written in a table as follows:
Three solutions of this equation can be written in a table as follows:
Hence,
we find that,
(ii) 9x + 3y + 12 = 0, 18x + 6y + 24 = 0
Comparing equation 9x + 3y + 12 = 0 with a
1
x + b
1
y + c
1
= 0and 7x + 6y – 9 = 0 with a
2
x + b
2
y + c
2
= 0,
We get,
Hence
We find that,
Hence, lines are coincident.
(iii) 6x − 3y + 10 = 0, 2x – y + 9 = 0
Comparing equation 6x − 3y + 10 = 0 with a
1
x + b
1
y + c
1
= 0and 7x + 6y – 9 = 0 with a
2
x + b
2
y + c
2
= 0,
We get,
Hence
We find that,
Hence,
lines are parallel to each other.
3. On comparing the ratios a
1
/a
2
,b
1
/b
2
and c
1
/c
2
, find out whether the following pair of linear equations are consistent, or inconsistent.
(i) 3x + 2y = 5, 2x − 3y = 8
(ii) 2x − 3y = 7, 4x − 6y = 9
(iii) 3x/2 + 5y/3 = 7, 9x − 10y = 14
(iv) 5x − 3y = 11, −10x + 6y = −22
Hence,
Therefore these linear equations will intersect at one point only and have only one possible solution.
And,pair of linear euations is consistent
(ii) 2x − 3y = 8, 4x − 6y = 9
Comparing equation 2x − 3y = 8 with a
1
x + b
1
y + c
1
= 0and 7x + 6y – 9 = 0 with a
2
x + b
2
y + c
2
= 0,
We get,
Hence,
Therefore these linear equations are parallel to each other and have no possible solution.in
And, pair of linear euations is inconsistent
(iii)
9x − 10y = 14
We get,
Hence,
Therefore, these linear equations will intersect each other at one point and have only one possible solution.
(iv) 5x − 3y = 11, −10x + 6y = −22
Comparing equation 5x − 3y = 11 with a
1
x + b
1
y + c
1
= 0and 7x + 6y – 9 = 0 with a
2
x + b
2
y + c
2
= 0,
We get,
Hence,
Therefore these pair of lines have infinite number of solutions
Hence,
(ii) x – y = 8, 3x − 3y = 16
We get,
Hence,
Therefore, these linear equations are intersecting each other at one point and thus have only one possible solution.
Hence,the pair of linear equations is inconsistent.
(iii) 2x + y = 6, 4x − 2y = 4
We get,
Hence,
Therefore, these linear equations are intersecting each other at one point and thus have only one possible solution.
Hence the pair of linear equations is consistent
(iv) 2x − 2y – 2 = 0, 4x − 4y – 5 = 0
We get,
Hence,
Therefore, these linear equations are parallel to each other and have no possible solution,
Hence,the pair of linear equations is inconsistent.
5. Half the perimeter of a rectangle garden, whose length is 4 m more than its width, is 36 m. Find the dimensions of the garden.
Hence, the graphic representation is as follows.
6. 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
The Second line such that it is intersecting the given line is
2x+4y-6=0
As,
(ii) Let the second line be equal to a
2
x + b
2
y + c
2
= 0,
parallel Lines:
For this Condition,
Hence,the second line can be 4x+6y-8=0
As,
(iii) Let the second line be equal to a
2
x + b
2
y + c
2
= 0,
Coincident lines: For coincident lines,
Hence,the second line can be 6x+9y-24=0
As,
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.
We can see from the graphs that points of intersection of the lines with the x–axis are (–1, 0), (2, 3) and (4, 0).
