Question 4: Which of the following pairs of linear equations are consistent/inconsistent? If consistent, obtain the solution graphically.

(a) `x + y = 5` and `2x + 2y = 10`

**Solution:** For the given pair of linear equations;

`(a_1)/(a_2)=1/2`

`(b_1)/(b_2)=1/2`

`(c_1)/(c_2)=(5)/(10)=1/2`

It is clear that;

`(a_1)/(a_2)= (b_1)/(b_2) =(c_1)/(c_2)`

Hence the given pair of linear equations is dependent and consistent. The lines for the equations will be coincident. This means there would be infinitely many solutions for the given pair of linear equations.

(b) `x – y = 8` and `3x – 3y = 16`

**Solution:** For the given pair of linear equations;

`(a_1)/(a_2)=1/3`

`(b_1)/(b_2)=1/3`

`(c_1)/(c_2)=(8)/(16)=1/2`

It is clear that;

`(a_1)/(a_2)= (b_1)/(b_2) ≠(c_1)/(c_2)`

Hence the given pair of linear equations is inconsistent.

(c) `2x + y – 6 = 0` and `4x – 2y – 4 = 0`

**Solution:** For the given pair of linear equations;

`(a_1)/(a_2)=2/4=1/2`

`(b_1)/(b_2)=(1)/(-2)`

It is clear that;

`(a_1)/(a_2)≠ (b_1)/(b_2)`

Hence the given pair of linear equations is consistent.

Following graph can be plotted with the given pair of linear equations;

Value of `x = 2` and that of `y = 2`

(d) `2x – 2y – 2 = 0` and `4x – 4y – 5 = 0`

**Solution:** For the given pair of linear equations;

`(a_1)/(a_2)=2/4=1/2`

`(b_1)/(b_2)=(-2)/(-4)=1/2`

`(c_1)/(c_2)=(-2)/(-5)=2/5`

It is clear that;

`(a_1)/(a_2)= (b_1)/(b_2) ≠(c_1)/(c_2)`

Hence the given pair of linear equations is inconsistent.

Question 5: Half the perimeter of a rectangular garden, whose length is 4 m more than its width, is 36 m. Find the dimensions of the garden.

**Solution:** Let us assume that width of garden = x and length = y. Then as per question;

x + 4 = y ……..(1)

x + y = 36 ………(2)

Substitution the value of y from equation (1) in equation (2) we get;

x + x + 4 = 36

Or, 2x + 4 = 36

Or, 2x = 32

Or, x = 16

Hence, y = x + 4 = 16 + 4 = 20

Hence, length = 20 m and breadth = 16 m

Question 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:

(a) Intersecting lines

**Solution:** For intersecting line, the linear equations should meet following condition:

`(a_1)/(a_2)≠ (b_1)/(b_2)`

For getting another equation to meet this criterion, multiply the coefficient of x with any number and multiply the coefficient of y with any other number. A possible equation can be as follows:

`4x + 4y – 8 = 0`

(b) Parallel lines

**Solution:** For parallel lines, the linear equations should meet following condition:

`(a_1)/(a_2)= (b_1)/(b_2) ≠(c_1)/(c_2)`

For getting another equation to meet this criterion, multiply the coefficients of x and y with the same number and multiply the constant term with any other number. A possible equation can be as follows:

`4x + 6y – 12 = 0`

(c) Coincident lines

**Solution:** For getting coincident lines, the equations should meet following condition;

`(a_1)/(a_2)= (b_1)/(b_2)=(c_1)/(c_2)`

For getting another equation to meet this criterion, multiply the whole equation with any number. A possible equation can be as follows:

`4x + 6y – 16 = 0`

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.

**Solution:** Following graph can be plotted with the given pair of linear equations:

The coordinates of vertices of the triangle are; (-1, 0), (2, 3) and (4, 0)

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