Write the equation of the line that passes through (3,2) and is parallel to 3x-2y=1
To find the equation of a line parallel to 3x-2y=1, we need to use the same slope as the given line.
First, we need to convert the given equation into slope-intercept form (y = mx + b), where m is the slope and b is the y-intercept.
Starting with the given equation:
3x - 2y = 1
We need to isolate y on one side of the equation, so let's subtract 3x from both sides:
-2y = -3x + 1
To isolate y, divide both sides of the equation by -2:
y = (3/2)x - 1/2
Now we can see that the slope of the given line is 3/2.
Since the line we want to find is parallel to this given line and passes through the point (3,2), we can use the point-slope form of a line, which is:
y - y1 = m(x - x1), where (x1, y1) is a point on the line and m is the slope.
In this case, (x1, y1) = (3,2) and m = 3/2.
Plugging in these values, we get:
y - 2 = (3/2)(x - 3)
Distributing the (3/2) on the right side:
y - 2 = (3/2)x - (3/2)(3)
Simplifying the right side:
y - 2 = (3/2)x - (9/2)
Adding 2 to both sides to isolate y:
y = (3/2)x - (9/2) + 2
Combining like terms on the right side:
y = (3/2)x - (9/2) + 4/2
Simplifying the right side:
y = (3/2)x - 5/2
So, the equation of the line that passes through (3,2) and is parallel to 3x - 2y = 1 is y = (3/2)x - 5/2.
To find the equation of a line parallel to the line 3x - 2y = 1 and passing through the point (3, 2), we can use the fact that parallel lines have the same slope.
First, let's rearrange the given equation into slope-intercept form (y = mx + b). Subtracting 3x from both sides gives us:
-2y = -3x + 1
Next, divide both sides by -2 to isolate y:
y = (3/2)x - 1/2
From the slope-intercept form, we can see that the slope of the given line is 3/2.
Since the line we are looking for is parallel to this line, it will have the same slope of 3/2.
Now, we can use the point-slope form of a line to find the equation. Given the point (3, 2) and the slope m = 3/2, the equation is:
y - y₁ = m(x - x₁)
Substituting the values of x₁, y₁, and m:
y - 2 = (3/2)(x - 3)
Next, we can simplify and rearrange the equation:
y - 2 = (3/2)x - (3/2)(3)
y - 2 = (3/2)x - 9/2
Finally, let's rewrite the equation in the standard form (Ax + By = C):
(3/2)x - y = 9/2 - 2
Multiplying every term by 2 to eliminate the fraction gives:
3x - 2y = 9 - 4
Simplifying further gives us the equation of the line:
3x - 2y = 5