Question

Draw the graph of the pair of linear equations given below and then state whether the lines are parallel or perpendicular:

x + 2y = 5 and 2x – y = 0

Answer 1

The given lines are perpendicular to each other.

1. Calculate the slopes

To determine the relationship between the lines, we first convert both equations into the slope-intercept form y = mx + c, where m represents the slope.

For equation 1 (x + 2y = 5):

2y = –x + 5

`y = -1/2 x + 5/2`

The slope m₁ = `-1/2`.

For equation 2 (2x – y = 0):

–y = –2x

y = 2x

The slope m₂ = 2.

2. Compare the slopes

Two lines are parallel if their slopes are equal (m₁ = m₂). They are perpendicular if the product of their slopesis –1 (m₁ · m₂ = –1).

Testing the product of the slopes:

 `m_1 * m_2 = (-1/2) * (2)`

m₁ · m₂ = –1

Since the product of the slopes is exactly –1, the lines intersect at a 90° angle.

3. Find the intersection point

By substituting y = 2x from the second equation into the first:

x + 2(2x) = 5

5x = 5

⇒ x = 1

Substituting x = 1 back into y = 2x:

y = 2(1)

y = 2

The lines intersect at the point (1, 2).

The lines are perpendicular because the product of their slopes is –1.