What is the Equation of a Parallel Line? | Simple Explanation for Class 10

S6-SA1-0266

What is the Equation of a Line Parallel to a Given Line?

Grade Level:

Class 10

AI/ML, Physics, Biotechnology, Space Technology, Chemistry, Engineering, Medicine

Definition

What is it?

The equation of a line parallel to a given line describes a new line that never intersects the original line. This new line will have the exact same 'steepness' or slope as the given line, but it will pass through a different point.

Simple Example

Quick Example

Imagine two parallel railway tracks in your city. Both tracks run in the same direction and never meet. If one track's path can be described by an equation, a parallel track will have an equation with the same direction (slope) but a different starting point (y-intercept).

Worked Example

Step-by-Step

Let's find the equation of a line parallel to 2x + 3y = 6 and passing through the point (1, 2).
---Step 1: Convert the given equation to the slope-intercept form (y = mx + c).
2x + 3y = 6
3y = -2x + 6
y = (-2/3)x + 2
---Step 2: Identify the slope (m) of the given line.
From y = (-2/3)x + 2, the slope m = -2/3.
---Step 3: Remember that parallel lines have the same slope.
So, the slope of our new parallel line will also be -2/3.
---Step 4: Use the point-slope form (y - y1 = m(x - x1)) with the new slope and the given point (1, 2).
y - 2 = (-2/3)(x - 1)
---Step 5: Simplify the equation to the slope-intercept form or standard form.
3(y - 2) = -2(x - 1)
3y - 6 = -2x + 2
2x + 3y = 8
---Answer: The equation of the line parallel to 2x + 3y = 6 and passing through (1, 2) is 2x + 3y = 8.

Why It Matters

Understanding parallel lines is key in many fields! In AI/ML, it helps create decision boundaries. Engineers use it to design bridges and buildings, ensuring structures are stable. Even in Space Technology, calculating parallel paths helps plan satellite orbits, making sure they don't collide.

Common Mistakes

MISTAKE: Using the negative reciprocal of the slope for parallel lines. | CORRECTION: Parallel lines have the EXACT SAME slope. The negative reciprocal is for perpendicular lines.

MISTAKE: Forgetting to use the given point (x1, y1) when finding the new equation. | CORRECTION: After finding the slope, always substitute the slope and the given point into the point-slope form (y - y1 = m(x - x1)) to find the correct y-intercept.

MISTAKE: Not converting the given equation to slope-intercept form (y = mx + c) first. | CORRECTION: Always rearrange the equation to isolate 'y' so you can clearly identify the slope 'm'.

Practice Questions

Try It Yourself

QUESTION: What is the slope of a line parallel to y = 5x - 7? | ANSWER: 5

QUESTION: Find the equation of a line parallel to y = 3x + 10 and passing through the point (0, 5). | ANSWER: y = 3x + 5

QUESTION: A line is parallel to 4x - 2y = 8 and passes through the point (-1, 3). Find its equation in standard form (Ax + By = C). | ANSWER: 2x - y = -5

MCQ

Quick Quiz

Which of these lines is parallel to the line y = -4x + 9?

y = 4x + 9

y = -4x - 2

y = (1/4)x + 5

y = -9x + 4

The Correct Answer Is:

Parallel lines have the same slope. The given line has a slope of -4. Option B also has a slope of -4, making it parallel. Other options have different slopes.

Real World Connection

In the Real World

Think about how your favourite mobile app like Google Maps works! When you search for 'parallel roads' or 'shortest path,' the algorithms behind it use concepts of parallel lines and slopes to map out routes and identify roads that run side-by-side, helping auto-rickshaw drivers or delivery partners navigate efficiently.

Key Vocabulary

Key Terms

SLOPE: The steepness of a line, represented by 'm' | Y-INTERCEPT: The point where a line crosses the y-axis, represented by 'c' | PARALLEL LINES: Lines that are always the same distance apart and never meet | EQUATION OF A LINE: A mathematical rule describing all points on a line

What's Next

What to Learn Next

Great job understanding parallel lines! Next, you should explore 'What is the Equation of a Line Perpendicular to a Given Line?' This builds on your knowledge of slopes and introduces a new relationship between lines, which is super important in geometry and real-world designs.