Graphing Lines: Equations, Perpendicularity, And Parallelism
Hey everyone! Today, we're diving into the world of lines in the Cartesian plane. We'll be working with equations, finding the slope, figuring out perpendicularity, and exploring parallelism. Sounds fun, right? Let's break down this math problem step-by-step, making it super easy to understand. We are going to find the equation of a line passing through two points, we will find the equation of a perpendicular line to the first one, and finally, we will find the equation of a parallel line to the first one. So, grab your pencils, and let's get started!
Finding the Equation of Line S
Alright, guys, our first mission is to find the equation of line S. This line passes through two points: (-1, 1) and (2, 7). To do this, we'll use the trusty slope-intercept form of a line, which is y = mx + b, where 'm' is the slope and 'b' is the y-intercept. So the first thing we need to find is the slope of the line, and that's the change in y over the change in x. It can be found using the following equation: m = (y2 - y1) / (x2 - x1).
Let's calculate the slope using the two points: (-1, 1) and (2, 7). We will assign the value of the first point to (x1, y1) and the second point to (x2, y2).
m = (7 - 1) / (2 - (-1)) m = 6 / 3 m = 2
So, the slope (m) of line S is 2. This means that for every 1 unit we move to the right on the x-axis, we move 2 units up on the y-axis. Great! Now, we have to find the value of b. To find 'b,' we'll plug the slope (m = 2) and one of the points (let's use (-1, 1)) into the slope-intercept form (y = mx + b) and solve for 'b'.
1 = 2 * (-1) + b 1 = -2 + b b = 3
Therefore, the y-intercept (b) is 3. The y-intercept is the point where the line crosses the y-axis. We now have everything we need to write the equation of line S. Substituting 'm' and 'b' into the slope-intercept form, we get: y = 2x + 3. That's it! We have the equation of line S. Congratulations! Now that we have the equation, let's graph it. To graph a line, we can find two points on the line and connect them. We already know two points: (-1, 1) and (2, 7). We can plot these points on the Cartesian plane and draw a straight line through them. This line represents the equation y = 2x + 3.
Finding the Equation of Line R (Perpendicular to S)
Alright, next up, we need to find the equation of line R. This line is perpendicular to line S and passes through the point (3, 3). Remember that perpendicular lines intersect at a 90-degree angle. A key concept here is that the slopes of perpendicular lines are negative reciprocals of each other. That means if the slope of line S is 'm', the slope of line R is '-1/m'. So now, we will find the value of the slope (m) of the line R.
We know that the slope of line S is 2 (from our previous calculations). Therefore, the slope of line R is -1/2 (the negative reciprocal of 2). Because (-1/2) * 2 = -1, and that's the rule! Now that we have the slope of line R (-1/2), and we know it passes through the point (3, 3), we can use the point-slope form of a line: y - y1 = m(x - x1), where 'm' is the slope and (x1, y1) is a point on the line. Let's substitute the slope m = -1/2, and the point (3,3) into the point-slope form:
y - 3 = -1/2 * (x - 3)
Now, let's simplify this equation into slope-intercept form (y = mx + b). First, distribute -1/2 to the parenthesis:
y - 3 = -1/2 * x + 3/2
Now, add 3 to both sides of the equation:
y = -1/2 * x + 3/2 + 3
y = -1/2 * x + 3/2 + 6/2 y = -1/2 * x + 9/2
So, the equation of line R is y = -1/2 * x + 9/2. Great job, guys! Now let's graph this line. To graph it, we can use the slope and the point (3, 3). From the point (3, 3), move 1 unit down (because the slope is negative, which means that for every 2 units to the right, we move 1 unit down) and 2 units to the right, and then plot the point. We can also use the y-intercept, which is the point (0, 9/2) or (0, 4.5), and connect the points to draw the line. It should form a 90-degree angle with line S.
Finding the Equation of a Line Parallel to S
Okay, in this last part of the problem, we need to find the equation of a line that is parallel to line S. Parallel lines never intersect; they have the same slope but different y-intercepts. So, since we know the slope of line S is 2, the slope of any line parallel to S will also be 2. Let's call this new line 'T'. The slope of line T is 2. The question does not specify a point that the line T passes through, so we are going to find the equation of a line with the same slope as S and the y-intercept of 0. That way, we're finding the equation of a line parallel to S (y = 2x + 3).
Since we're going to create a line with the y-intercept of 0, the equation would be:
y = 2x + 0 y = 2x
So, the equation of line T is y = 2x. To graph this line, we can find two points. The first one is (0, 0), and the second one can be (1, 2). Plot the points, and draw the line. We can see that the line is parallel to the line S. Great job!
Summary and Key Takeaways
Let's recap what we've learned, guys! We started with two points and found the equation of a line (line S), using the slope-intercept form. We calculated the slope and found the y-intercept, then we graphed it. Next, we found the equation of a line perpendicular to line S (line R). Remember that the slopes of perpendicular lines are negative reciprocals of each other. We used the point-slope form and converted it into slope-intercept form to find the equation and then graphed it. Finally, we found the equation of a line parallel to line S (line T). Parallel lines have the same slope. By finding the equation, we saw how parallel lines behave and graphed it.
Key Takeaways:
- The slope-intercept form of a line is y = mx + b. m represents the slope, and b represents the y-intercept.
- The slope (m) can be calculated using the formula: m = (y2 - y1) / (x2 - x1).
- Perpendicular lines have slopes that are negative reciprocals of each other.
- Parallel lines have the same slope.
Conclusion
And that's a wrap! You guys did an amazing job today. We've successfully navigated through equations, slopes, perpendicularity, and parallelism. Remember, practice makes perfect. The more you work with these concepts, the easier they'll become. Keep up the great work, and don't be afraid to ask questions. Until next time, keep exploring the fascinating world of mathematics! Bye!