Need help with two questions. I've been stuck on them for a while now.
1) Find a general form of an equation of the line through the point A that satisfies the given condition.
A(−2, 5) slope 3/4
2)Find a general form of an equation of the line through the point A that satisfies the given condition.
A(3, −6) parallel to the line 7x − 3y = 6
Thanks!
I'd be happy to help you with your questions!
1) To find the equation of a line through a given point with a known slope, you can use the point-slope form of a linear equation. The point-slope form is given by:
y - y1 = m(x - x1)
where (x1, y1) represents the coordinates of the point and m represents the slope.
In this case, the point A has coordinates (-2, 5) and the slope is 3/4. Plugging these values into the point-slope form, we have:
y - 5 = (3/4)(x - (-2))
Simplifying this equation, we get:
y - 5 = (3/4)(x + 2)
To convert this to the general form of a linear equation (Ax + By + C = 0), we can multiply through by 4 to clear the fraction:
4(y - 5) = 3(x + 2)
Expanding both sides of the equation, we get:
4y - 20 = 3x + 6
Rearranging the terms, we have:
3x - 4y + 26 = 0
So, the general form of the equation of the line through point A(-2, 5) with a slope of 3/4 is 3x - 4y + 26 = 0.
2) To find the equation of a line parallel to a given line, you need to note that parallel lines have the same slope. The given line has an equation of 7x - 3y = 6. To determine the slope of this line, we can rearrange the equation into slope-intercept form (y = mx + b), where m represents the slope:
-3y = -7x + 6
y = (7/3)x - 2
From the slope-intercept form, we can see that the slope of the given line is 7/3.
Since the line we're looking for is parallel to the given line, it will also have a slope of 7/3.
Next, we can use the point-slope form of a linear equation, as we did in the first question, to find the equation of the line through point A(3, -6) with a slope of 7/3:
y - (-6) = (7/3)(x - 3)
Simplifying this equation, we get:
y + 6 = (7/3)(x - 3)
To convert this to the general form of a linear equation, we can multiply through by 3 to clear the fraction:
3(y + 6) = 7(x - 3)
Expanding both sides of the equation, we get:
3y + 18 = 7x - 21
Rearranging the terms, we have:
7x - 3y + 39 = 0
So, the general form of the equation of the line through point A(3, -6) parallel to the line 7x - 3y = 6 is 7x - 3y + 39 = 0.