Distance Between the Parallel Lines 2x + 5y = 7 and 2x + 5y = 15

Understanding the Given Parallel Lines

We are given two straight lines:

2x + 5y = 7

and

2x + 5y = 15

The first important observation is that both equations have exactly the same coefficients of x and y. In both equations, the coefficient of x is 2 and the coefficient of y is 5. Therefore, the two lines have the same slope and are parallel to each other.

Since parallel lines never intersect, the distance between them is defined as the length of the shortest perpendicular segment joining the two lines. Coordinate geometry provides a direct formula to calculate this perpendicular distance.

Formula for the Distance Between Two Parallel Lines

If two parallel lines are written in the standard forms:

Ax + By + C₁ = 0

and

Ax + By + C₂ = 0,

then the perpendicular distance between the two parallel lines is given by:

d = |C₂ − C₁| / √(A² + B²)

Here, A and B are the common coefficients of x and y, while C₁ and C₂ are the constant terms of the two line equations after they have been written in standard form.

Step-by-Step Solution

Step 1: Convert Both Equations into Standard Form

The first given line is:

2x + 5y = 7

Bringing 7 to the left-hand side gives:

2x + 5y − 7 = 0

The second given line is:

2x + 5y = 15

Bringing 15 to the left-hand side gives:

2x + 5y − 15 = 0

Comparing these equations with the standard forms Ax + By + C₁ = 0 and Ax + By + C₂ = 0, we obtain:

A = 2, B = 5, C₁ = −7, and C₂ = −15

Step 2: Substitute the Values into the Distance Formula

Using the formula for the distance between two parallel lines:

d = |C₂ − C₁| / √(A² + B²)

Substituting A = 2, B = 5, C₁ = −7, and C₂ = −15:

d = |−15 − (−7)| / √(2² + 5²)

Simplifying the numerator:

|−15 + 7| = |−8| = 8

Simplifying the denominator:

√(2² + 5²) = √(4 + 25) = √29

Therefore:

d = 8 / √29

Step 3: Calculate the Numerical Value

The approximate value of √29 is:

√29 ≈ 5.385

Therefore:

d = 8 / 5.385

d ≈ 1.4856

The question asks for the answer rounded off to two decimal places. Therefore:

d ≈ 1.49

Why the Absolute Value Is Important

The distance between two geometric objects can never be negative. Depending on the order in which C₁ and C₂ are subtracted, the numerator may initially become negative. The absolute value sign ensures that the final distance is always positive.

For example, in this problem, −15 − (−7) gives −8. Taking the absolute value changes this to 8, which represents the actual positive distance between the two parallel lines.

Geometrical Interpretation of the Answer

The equations 2x + 5y = 7 and 2x + 5y = 15 represent two distinct parallel straight lines. Since the coefficients of x and y are identical, their directions are exactly the same, but their different constant values place them at different positions in the coordinate plane.

The calculated value 1.49 represents the shortest perpendicular distance between these two lines. This distance remains constant at every point because the lines are parallel.

Final Answer

The distance between the parallel lines 2x + 5y = 7 and 2x + 5y = 15 is 1.49 units.