Equation Of A Parallel Line: Step-by-Step Solution
Have you ever wondered how to find the equation of a line that runs perfectly parallel to another? It's a common problem in mathematics, especially in algebra and geometry. In this guide, we'll break down the process step-by-step, using a specific example to illustrate each concept. So, let's dive in and unravel the mystery of parallel lines!
Understanding the Basics of Parallel Lines
Before we jump into solving the problem, let's make sure we're all on the same page about what parallel lines are and their key properties. Parallel lines are lines in a plane that never intersect. They maintain a constant distance from each other, no matter how far they extend. Think of train tracks running side by side – that's a perfect example of parallel lines in the real world.
A crucial property of parallel lines is that they have the same slope. The slope of a line represents its steepness and direction. It's often denoted by the letter 'm' and calculated as the "rise over run" between any two points on the line. If two lines have the same slope, they are either parallel or the same line (if they also share the same y-intercept). This concept is the foundation for finding the equation of a parallel line.
Slope-Intercept Form: The Key to Our Solution
The slope-intercept form of a linear equation is a powerful tool for understanding and manipulating lines. It's expressed as:
- y = mx + b
Where:
- 'y' represents the vertical coordinate
- 'x' represents the horizontal coordinate
- 'm' is the slope of the line
- 'b' is the y-intercept (the point where the line crosses the y-axis)
This form is incredibly useful because it directly reveals the slope ('m') and the y-intercept ('b') of the line. Knowing these two values allows us to easily graph the line and write its equation. In our problem, we'll use the slope-intercept form to determine the equation of the parallel line.
Problem Statement: Finding the Equation of Line q
Now, let's tackle the specific problem we're faced with. We're given the following information:
- Line p has the equation: y = -5/6x - 5
- Line q is parallel to line p
- Line q passes through the point (9, -4)
Our goal is to find the equation of line q in slope-intercept form (y = mx + b). To achieve this, we'll use the properties of parallel lines and the given information to determine the slope and y-intercept of line q.
Step-by-Step Solution: Unraveling the Equation
Let's break down the solution into manageable steps:
Step 1: Identify the Slope of Line p
The equation of line p is given in slope-intercept form: y = -5/6x - 5. By comparing this to the general form (y = mx + b), we can directly identify the slope of line p.
- The coefficient of x is the slope, so the slope of line p is -5/6.
Step 2: Determine the Slope of Line q
This is where the key property of parallel lines comes into play. Since line q is parallel to line p, it must have the same slope. Therefore:
- The slope of line q (m) is also -5/6.
Step 3: Use the Point-Slope Form
Now that we know the slope of line q and a point it passes through (9, -4), we can use the point-slope form of a linear equation to find its equation. The point-slope form is:
- y - y₁ = m(x - x₁)
Where:
- (x₁, y₁) is a point on the line
- m is the slope of the line
Let's plug in the values we know:
- x₁ = 9
- y₁ = -4
- m = -5/6
So, our equation becomes:
- y - (-4) = -5/6(x - 9)
Step 4: Simplify the Equation
Let's simplify the equation we obtained in the previous step:
- y + 4 = -5/6(x - 9)
First, distribute the -5/6 on the right side:
-
y + 4 = -5/6x + (5/6 * 9)
-
y + 4 = -5/6x + 45/6
Simplify the fraction 45/6:
- y + 4 = -5/6x + 15/2
Step 5: Convert to Slope-Intercept Form
To get the equation in slope-intercept form (y = mx + b), we need to isolate y. Subtract 4 from both sides of the equation:
- y = -5/6x + 15/2 - 4
To combine the constants, we need a common denominator. Convert 4 to a fraction with a denominator of 2:
- y = -5/6x + 15/2 - 8/2
Now, combine the fractions:
- y = -5/6x + 7/2
Step 6: The Final Equation
We've successfully transformed the equation into slope-intercept form. The equation of line q is:
- y = -5/6x + 7/2
This equation tells us that line q has a slope of -5/6 and a y-intercept of 7/2.
Conclusion: Mastering Parallel Lines
Congratulations! You've successfully navigated the process of finding the equation of a parallel line. By understanding the properties of parallel lines, the slope-intercept form, and the point-slope form, you can confidently tackle similar problems. Remember, the key is to break down the problem into manageable steps and apply the relevant concepts.
This step-by-step approach not only helps in solving mathematical problems but also enhances your understanding of the underlying principles. So, keep practicing and exploring the fascinating world of mathematics!
For further exploration on linear equations and parallel lines, you can visit Khan Academy's Linear Equations section. This resource provides comprehensive lessons, practice exercises, and videos to deepen your understanding.
