Question

Solve the following system of equations graphically:

x + 3y = 6 and 2x – 3y = 12

Also, find the area of the triangle formed by the lines x + 3y = 6, x = 0 and y = 0.

Answer 1

1. Find intercepts and plot equations

To solve the system graphically, find at least two points for each line:

For x + 3y = 6:

If x = 0, then 3y = 6

⇒ y = 2

Point: (0, 2)

If y = 0, then x = 6

Point: (6, 0)

For 2x – 3y = 12:

If x = 0, then –3y = 12

⇒ y = –4

Point: (0, –4)

If y = 0, then 2x = 12

⇒ x = 6

Point: (6, 0)

2. Identify the intersection

Plotting these lines on a coordinate plane shows they intersect at the point where their coordinates match.

Both lines pass through the point (6, 0)

Therefore, the solution to the system is x = 6 and y = 0.

3. Calculate the triangle area

The triangle formed by x + 3y = 6, x = 0 (y-axis) and y = 0 (x-axis) has the following vertices:

1. Origin: (0, 0)
2. y-intercept of x + 3y = 6: (0, 2)
3. x-intercept of x + 3y = 6: (6, 0)

Using the area formula Area = `1/2 xx "base" xx "height"`:

Base (along x-axis) = 6 units

Height (along y-axis) = 2 units

Area = `1/2 xx 6 xx 2 = 6` square units.

The graphical solution is (6, 0) and the area of the triangle formed by x + 3y = 6, x = 0 and y = 0 is 6 square units.