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

We are given the linear equation 2x + 3y - 8 = 0. We need to write another linear equation in two variables such that the geometrical representation of the pair of lines formed is one of the following:

(i) Intersecting lines

For the lines to be intersecting, they must have different slopes. The given equation can be written in slope-intercept form as:

2x + 3y - 8 = 0
=> 3y = -2x + 8
=> y = (-2/3)x + 8/3

The slope of this line is -2/3. To get intersecting lines, we need another line with a different slope. For example, the equation of the second line can be:

4x + 3y - 6 = 0

Let's check its slope:

4x + 3y - 6 = 0
=> 3y = -4x + 6
=> y = (-4/3)x + 2

The slope of this line is -4/3, which is different from -2/3. Therefore, the lines represented by 2x + 3y - 8 = 0 and 4x + 3y - 6 = 0 will intersect.

(ii) Parallel lines

For the lines to be parallel, they must have the same slope but different y-intercepts. From the previous step, we know that the slope of the given line is -2/3. To create a parallel line, we can keep the slope the same and change the constant term. For example:

2x + 3y - 5 = 0

Let's check its slope:

2x + 3y - 5 = 0
=> 3y = -2x + 5
=> y = (-2/3)x + 5/3

The slope of this line is also -2/3, which is the same as the given line. Therefore, the lines represented by 2x + 3y - 8 = 0 and 2x + 3y - 5 = 0 will be parallel.

(iii) Coincident lines

For the lines to be coincident, they must have the same slope and the same y-intercept. This means the equations must be identical. So, we can simply multiply the given equation by a constant. For example:

4x + 6y - 16 = 0

Let's check if this is coincident with the given equation. Dividing this by 2:

(4x + 6y - 16) / 2 = 2x + 3y - 8 = 0

So, the line represented by 4x + 6y - 16 = 0 is the same as the original equation 2x + 3y - 8 = 0. Therefore, the lines are coincident.

Summary:

(i) For intersecting lines: 4x + 3y - 6 = 0
(ii) For parallel lines: 2x + 3y - 5 = 0
(iii) For coincident lines: 4x + 6y - 16 = 0