Final Answer:
The equation of a line in slope-intercept form that passes through (3, -2) and is perpendicular to the line x−4y=−3 is (b) y=1/4x−11/4.
Step-by-step explanation:
To find the equation of a line perpendicular to the given line x-4y=-3 and passing through the point (3, -2), we first need to determine the slope of the original line. The given line can be rewritten in the slope-intercept form y = mx + b, where m is the slope and b is the y-intercept. Rearranging the equation x-4y=-3 in this form, we get y = 1/4x + 3/4. The slope m of this line is 1/4.
Since the required line is perpendicular, the negative reciprocal of 1/4 is -4. Now, we use the point-slope form y - y₁ = m(x - x₁), where (x₁, y₁) is the given point (3, -2) and m is the slope. Substituting the values, we get y - (-2) = -4(x - 3). Simplifying this equation leads to y = -4x + 10.
Further, we can express the equation in the slope-intercept form by rearranging the terms, resulting in y = 1/4x - 11/4. Thus, the correct answer is option (b) y=1/4x−11/4.