Find the equation of the line parallel to the line 2x +3y = -18 that passes through the point (2, -3) in the slope-intercept form.
start with the point-slope form, since that is what they gave you.
The slope you want is -2/3, so the line you want is
y+3 = -2/3 (x-2)
Now just massage that into y=mx+b form
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Why did the line go on a parallel quest? Because it didn't want to be perpendicularity challenged!
Anyhow, to find the equation of a line parallel to 2x + 3y = -18, we need to determine the slope of the given line.
Let's rearrange the given equation to the slope-intercept form, y = mx + b:
2x + 3y = -18
3y = -2x - 18
y = (-2/3)x - 6
The slope of this line is -2/3.
Since the line we're looking for is parallel, it will have the same slope of -2/3.
Now, we can use the point-slope form of a line to find the equation:
y - y₁ = m(x - x₁)
Plugging in the values from the point (2, -3):
y - (-3) = (-2/3)(x - 2)
y + 3 = (-2/3)(x - 2)
Now, let's simplify it further:
y + 3 = (-2/3)x + (4/3)
y = (-2/3)x + (4/3) - 3
y = (-2/3)x + (4/3) - (9/3)
y = (-2/3)x - (5/3)
Thus, the equation of the line parallel to 2x + 3y = -18, which passes through the point (2, -3), is y = (-2/3)x - (5/3) in the slope-intercept form.
To find the equation of a line parallel to a given line, we need to determine the slope of the given line and then use that slope to write the equation of the new line.
The equation of the given line is 2x + 3y = -18. First, let's rewrite the equation in slope-intercept form (y = mx + b), where m is the slope and b is the y-intercept.
To do that, isolate y by subtracting 2x from both sides of the equation:
3y = -2x - 18
Divide both sides of the equation by 3 to solve for y:
y = (-2/3)x - 6
Now, we can see that the slope of the given line is -2/3. Since we want to find the equation of a line parallel to this line, the new line will have the same slope.
We are also given that the new line passes through the point (2, -3). We can use the slope-intercept form to write the equation of the new line.
Using the point-slope form (y - y1 = m(x - x1)), we substitute the values of x1, y1, and m into the equation:
y - (-3) = (-2/3)(x - 2)
Simplifying the equation further:
y + 3 = (-2/3)x + 4/3
Subtract 3 from both sides of the equation:
y = (-2/3)x + 4/3 - 3
Convert 4/3 to its decimal form:
y = (-2/3)x + 4/3 -9/3
Combine fractions:
y = (-2/3)x - 5/3
Therefore, the equation of the line parallel to 2x + 3y = -18 that passes through the point (2, -3) in slope-intercept form is y = (-2/3)x - 5/3.