Question

Solve the following given system of equations graphically and find the vertices and area of the triangle formed by these lines and the x-axis:

x – y + 3 = 0, 2x + 3y – 4 = 0

Answer 1

1. Find points for Line 1

Express the first equation x – y + 3 = 0 in terms of y:

y = x + 3

Find at least two points to plot the line:

If x = 0 ⇒ y = 3 ⇒ (0, 3)

If y = 0 ⇒ x = –3 ⇒ (–3, 0)

2. Find points for Line 2

Express the second equation 2x + 3y – 4 = 0 in terms of y:

`y = (4 - 2x)/3`

Find at least two points to plot the line:

If x = 2 ⇒ y = `(4 - 4)/3` = 0 ⇒ (2, 0)

If x = –1 ⇒ y = `(4 - 2(-1))/3` = 2 ⇒ (–1, 2)

3. Determine the vertices

Plotting both lines reveals that they intersect at the point (–1, 2).

The triangle is formed by the two lines and the x-axis (where y = 0). The three vertices are;

Intersection point of the two lines: A(–1, 2)

x-intercept of the first line: B(–3, 0)

y-intercept of the second line: C(2, 0)

4. Calculate the area

The base of the triangle lies along the x-axis between x = –3 and x = 2:

Base = 2 – (–3) = 5 units

The height of the triangle is given by the y-coordinate of the intersection point A:

Height = 2 units

Using the standard triangle area formula:

Area = `1/2 xx "base" xx "height"`

= `1/2 xx 5 xx 2`

= 5 sq. units