Maths Solve the Following Question.(5 Marks)

Consider the line whose equation is 5x – 3y = 0. The constant term is zero, therefore this line is passing through the origin. ∴ one point on the line is the origin O = (0, 0). To find the other point, we can give any value of x and get the corresponding value of y. Put x = 3, we get 15 – 3y = 0, i.e. y = 5 ∴ A ≡ (3, 5) is another point on the line. Draw the line OA. To find the solution set, we cannot check 0(0, 0), as it is already on the line. We can check any other point which is not on the line. Let us check the point (1, -1). When x = 1, y = -1 then 5x – 3y = 5 + 3 = 8 which is neither less nor equal to zero. ∴ 5x – 3y ≰ 0 in this case. Hence (1, -1) will not lie in the required region. Therefore, the required region is the upper side which is shaded in the graph.

This is the solution set of 5x – 3y ≤ 0.

Consider the line whose equation is 3x + 2y = 0. The constant term is zero, therefore this line is passing through the origin. ∴ one point on the line is O ≡ (0, 0). To find the another point, we can give any value of x and get the corresponding value of y. Put x = 2, we get 6 + 2y = 0 i.e. y = – 3 ∴ A = (2, -3) is another point on the line. Draw the line OA. To find the solution set, we cannot check (0, 0) as it is already on the line. We can check any other point which is not on the line. Let us check the point (1, 1)

When x = 1, y = 1, then 3x + 2y = 3 + 2 = 5 which is greater than zero. ∴ 3x + 2y > 0 in this case. Hence (1, 1) lies in the required region. Therefore, the required region is the upper side which is shaded in the graph. This is the solution set of 3x + 2y ≥ 0.

Consider the line whose equation is 2x – 5y = 10. To find the points of intersection of this line with the coordinate axes. Put y = 0, we get 2x = 10, i.e. x = 5. ∴ A = (5, 0) is a point on the line. Put x = 0, we get -5y = 10, i.e. y = -2 ∴ B = (0, -2) is another point on the line.

Draw the line AB joining these points. This line J divide the plane in two parts. 1. Origin side 2. Non-origin side To find the solution set, we have to check the position of the origin (0, 0) with respect to the line. When x = 0, y = 0, then 2x – 5y = 0 which is neither greater nor equal to 10. ∴ 2x – 5y ≱ 10 in this case. Hence (0, 0) will not lie in the required region. Therefore, the given inequality is the non-origin side, which is shaded in the graph. This is the solution set of 2x – 5y ≥ 10.

Consider the line whose equation is x + 2y = 6. To find the points of intersection of this line with the coordinate axes. Put y = 0, we get x = 6. ∴ A = (6, 0) is a point on the line. Put x = 0, we get 2y = 6, i.e. y = 3 ∴ B = (0, 3) is another point on the line.

Draw the line AB joining these points. This line divide the line into two parts. 1. Origin side 2. Non-origin side To find the solution set, we have to check the position of the origin (0, 0) with respect to the line. When x = 0, y = 0, then x + 2y = 0 which is less than 6. ∴ x + 2y ≤ 6 in this case. Hence, origin lies in the required region. Therefore, the given inequality is the origin side which is shaded in the graph. This is the solution set of x + 2y ≤ 6.