Write An Equation Of The Line That Passes Through The Point (9, 4) And Is Parallel To The Line \[$y - X = 3\$\].

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Introduction

In mathematics, the concept of parallel lines is crucial in understanding various geometric and algebraic relationships. When two lines are parallel, they never intersect, and their slopes are equal. In this article, we will explore how to write an equation of a line that passes through a given point and is parallel to another line.

Understanding the Problem

Given a point (9, 4) and a line y - x = 3, we need to find the equation of a line that passes through the point (9, 4) and is parallel to the line y - x = 3. To solve this problem, we will first rewrite the given line in slope-intercept form (y = mx + b), where m is the slope and b is the y-intercept.

Rewriting the Given Line

The given line is y - x = 3. To rewrite it in slope-intercept form, we need to isolate y.

y - x = 3
y = x + 3

Now, we can see that the slope of the given line is 1, and the y-intercept is 3.

Finding the Slope of the Parallel Line

Since the line we are looking for is parallel to the given line, it must have the same slope. Therefore, the slope of the parallel line is also 1.

Writing the Equation of the Parallel Line

Now that we have the slope (m = 1) and a point (9, 4) that the line passes through, we can use the point-slope form of a line (y - y1 = m(x - x1)) to write the equation of the parallel line.

y - 4 = 1(x - 9)
y - 4 = x - 9
y = x - 5

Conclusion

In this article, we have learned how to write an equation of a line that passes through a given point and is parallel to another line. We first rewrote the given line in slope-intercept form to find its slope and y-intercept. Then, we used the point-slope form of a line to write the equation of the parallel line. The final equation of the parallel line is y = x - 5.

Example Use Cases

  1. Architecture: When designing buildings, architects need to ensure that the lines of the building's walls are parallel to each other. By using the concept of parallel lines, architects can create a stable and aesthetically pleasing structure.
  2. Engineering: In engineering, parallel lines are used to design and build various structures, such as bridges and roads. By understanding the concept of parallel lines, engineers can create safe and efficient transportation systems.
  3. Computer Graphics: In computer graphics, parallel lines are used to create 3D models and animations. By using the concept of parallel lines, artists and designers can create realistic and engaging visual effects.

Tips and Tricks

  1. Use the Point-Slope Form: When writing the equation of a line that passes through a given point and is parallel to another line, use the point-slope form (y - y1 = m(x - x1)) to ensure accuracy.
  2. Check the Slope: Before writing the equation of the parallel line, check that the slope of the given line is the same as the slope of the parallel line.
  3. Use Real-World Examples: When teaching or learning about parallel lines, use real-world examples to illustrate the concept. This can help students understand the practical applications of parallel lines.

Common Mistakes

  1. Incorrect Slope: When writing the equation of the parallel line, make sure to use the correct slope. If the slope is incorrect, the equation of the parallel line will be incorrect.
  2. Incorrect Point: When writing the equation of the parallel line, make sure to use the correct point. If the point is incorrect, the equation of the parallel line will be incorrect.
  3. Not Checking the Slope: Before writing the equation of the parallel line, make sure to check that the slope of the given line is the same as the slope of the parallel line. If the slope is not the same, the equation of the parallel line will be incorrect.

Conclusion

Q: What is the definition of parallel lines?

A: Parallel lines are lines that lie in the same plane and never intersect, no matter how far they are extended. They have the same slope and are equidistant from each other.

Q: How do I determine if two lines are parallel?

A: To determine if two lines are parallel, you can use the following methods:

  • Check if the lines have the same slope.
  • Check if the lines are equidistant from each other.
  • Check if the lines intersect at any point.

Q: What is the equation of a line that passes through a given point and is parallel to another line?

A: The equation of a line that passes through a given point (x1, y1) and is parallel to another line with slope m is given by:

y - y1 = m(x - x1)

Q: How do I find the equation of a line that passes through a given point and is parallel to another line?

A: To find the equation of a line that passes through a given point (x1, y1) and is parallel to another line with slope m, follow these steps:

  1. Write the equation of the given line in slope-intercept form (y = mx + b).
  2. Identify the slope (m) of the given line.
  3. Use the point-slope form (y - y1 = m(x - x1)) to write the equation of the parallel line.
  4. Simplify the equation to obtain the final form.

Q: What is the difference between parallel lines and perpendicular lines?

A: Parallel lines are lines that lie in the same plane and never intersect, while perpendicular lines are lines that intersect at a right angle (90 degrees). Perpendicular lines have slopes that are negative reciprocals of each other.

Q: How do I find the equation of a line that is perpendicular to another line?

A: To find the equation of a line that is perpendicular to another line with slope m, follow these steps:

  1. Identify the slope (m) of the given line.
  2. Find the negative reciprocal of the slope (m' = -1/m).
  3. Use the point-slope form (y - y1 = m'(x - x1)) to write the equation of the perpendicular line.
  4. Simplify the equation to obtain the final form.

Q: What are some real-world applications of parallel lines?

A: Parallel lines have many real-world applications, including:

  • Architecture: Parallel lines are used in building design to create stable and aesthetically pleasing structures.
  • Engineering: Parallel lines are used in engineering to design and build various structures, such as bridges and roads.
  • Computer Graphics: Parallel lines are used in computer graphics to create 3D models and animations.

Q: What are some common mistakes to avoid when working with parallel lines?

A: Some common mistakes to avoid when working with parallel lines include:

  • Incorrect slope: Make sure to use the correct slope when writing the equation of a parallel line.
  • Incorrect point: Make sure to use the correct point when writing the equation of a parallel line.
  • Not checking the slope: Make sure to check that the slope of the given line is the same as the slope of the parallel line.

Conclusion

In conclusion, parallel lines are an essential concept in mathematics and have many real-world applications. By understanding the definition, properties, and equations of parallel lines, you can solve problems and create designs with ease. Remember to avoid common mistakes and use the correct slope and point when working with parallel lines. With practice and patience, you will become proficient in working with parallel lines.