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What is the Equation of a Line Passing Through a Point and Parallel to another Line?
Grade Level:
Class 10
AI/ML, Physics, Biotechnology, Space Technology, Chemistry, Engineering, Medicine
Definition
What is it?
The equation of a line passing through a specific point and parallel to another line describes a new line that never intersects the given line, maintaining the same 'slant' or slope. Since parallel lines have the same slope, we use the slope of the given line and the coordinates of the given point to find the equation of the new line.
Simple Example
Quick Example
Imagine two parallel railway tracks in your city. If you know the path (equation) of one track and a specific point where a new track needs to start, but it must run exactly parallel to the first, you can find the equation for this new track. Both tracks will have the same 'slope' or direction.
Worked Example
Step-by-Step
Find the equation of a line passing through the point (3, 5) and parallel to the line y = 2x + 1.
STEP 1: Identify the slope of the given line. The given line is y = 2x + 1. It is in the form y = mx + c, where 'm' is the slope. So, the slope (m) of this line is 2.
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STEP 2: Since the new line is parallel to the given line, its slope will be the same. So, the slope of our new line (m_new) is also 2.
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STEP 3: Use the point-slope form of a linear equation: y - y1 = m(x - x1). Here, (x1, y1) is the point (3, 5) and m is 2.
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STEP 4: Substitute the values into the point-slope form: y - 5 = 2(x - 3).
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STEP 5: Simplify the equation: y - 5 = 2x - 6.
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STEP 6: Rearrange to the standard form (y = mx + c): y = 2x - 6 + 5.
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STEP 7: Final equation: y = 2x - 1.
ANSWER: The equation of the line is y = 2x - 1.
Why It Matters
Understanding parallel lines helps engineers design roads and bridges, ensuring structures are stable. In AI/ML, similar concepts are used to train models for tasks like image recognition. Doctors use it for precise medical imaging, and space scientists use it to plot satellite orbits, making sure they don't collide!
Common Mistakes
MISTAKE: Using the y-intercept of the given line for the new line. | CORRECTION: Only the slope (m) is the same for parallel lines. The y-intercept (c) will be different for the new line unless it's the exact same line.
MISTAKE: Confusing parallel with perpendicular lines. | CORRECTION: Parallel lines have the SAME slope. Perpendicular lines have slopes that are negative reciprocals of each other (m1 * m2 = -1).
MISTAKE: Incorrectly applying the point-slope formula (y - y1 = m(x - x1)). | CORRECTION: Double-check that you are substituting the correct x1, y1 from the given point and the correct slope 'm' into the formula.
Practice Questions
Try It Yourself
QUESTION: Find the equation of a line passing through (1, 4) and parallel to y = 3x - 2. | ANSWER: y = 3x + 1
QUESTION: A line passes through (-2, 0) and is parallel to the line 2x + y = 7. Find its equation. | ANSWER: y = -2x - 4
QUESTION: The cost of mobile data (y) is Rs. 10 per GB (x) plus a fixed charge of Rs. 50, so y = 10x + 50. If a new plan offers data at the same rate but has a fixed charge of Rs. 30, what is the equation for this new plan? Is it parallel to the old plan's equation? | ANSWER: y = 10x + 30. Yes, it is parallel because the slope (rate per GB) is the same (10).
MCQ
Quick Quiz
Which of the following lines is parallel to y = 4x + 7 and passes through the point (1, 10)?
y = 4x + 6
y = -4x + 14
y = x + 9
y = 4x + 10
The Correct Answer Is:
A
The given line has a slope of 4. A parallel line must also have a slope of 4. Option A (y = 4x + 6) has a slope of 4. When you substitute (1, 10) into y = 4x + 6, you get 10 = 4(1) + 6, which is 10 = 10, so it passes through the point.
Real World Connection
In the Real World
When city planners design new roads, they often need to create parallel lanes or service roads that maintain a consistent distance from existing main roads. Using this concept, they can calculate the exact path (equation) for the new parallel road, ensuring smooth traffic flow and efficient land use, much like how ISRO engineers plot parallel paths for satellites.
Key Vocabulary
Key Terms
SLOPE: The 'steepness' or 'slant' of a line, represented by 'm' in y = mx + c. | PARALLEL LINES: Two lines that never intersect and have the same slope. | POINT-SLOPE FORM: A way to write the equation of a line: y - y1 = m(x - x1). | Y-INTERCEPT: The point where a line crosses the y-axis, represented by 'c' in y = mx + c.
What's Next
What to Learn Next
Great job understanding parallel lines! Next, you should learn about the 'Equation of a Line Passing Through a Point and Perpendicular to another Line'. This will teach you about lines that meet at a perfect right angle, building on your knowledge of slopes in a new way.
