Write your answer in standard form. Show all your work if you want to be eligible for partial credit.
Write the equation of a line perpendicular to the line y = (2/3)x - 7 and passing through the point (4,-2).
The slope of the line y = (2/3)x - 7 is 2/3. The slope of a line perpendicular to this line must be its negative reciprocal, or -3/2.
Using the point-slope form to find the other line:
y - y1 = m(x - x1)
y - (-2) = -3/2(x - 4)
y + 2 = (-3/2)x + 6
y = (-3/2)x + 4
In standard form:
(3/2)x + y = 4
To find the equation of a line perpendicular to another line, you need to determine the slope of the original line and then find the negative reciprocal of that slope.
In this case, the equation of the original line is y = (2/3)x - 7, which means the slope of that line is 2/3. To find the slope of a line perpendicular to this, you need to take the negative reciprocal of 2/3. The negative reciprocal is -3/2.
Once you have the slope of the perpendicular line, you can use the point-slope form to find the equation of the line. The point-slope form is given by:
y - y1 = m(x - x1)
where (x1, y1) is a point on the line and m is the slope.
In this case, the point given is (4, -2), so we can substitute these values into the equation:
y - (-2) = -3/2(x - 4)
Simplifying this equation, we get:
y + 2 = (-3/2)x + 6
Next, we can isolate y on one side of the equation:
y = (-3/2)x + 4
Finally, we can rewrite the equation in standard form by multiplying each term by 2 to eliminate fractions:
2y = -3x + 8
Rearranging the terms in standard form, we get:
3x + 2y = 8
So, the equation of a line perpendicular to y = (2/3)x - 7 and passing through the point (4, -2) is 3x + 2y = 8 in standard form.