Find the equation of the line that passes through the point (4, -1) and is perpe

Find the equation of the line that passes through the point (4, -1) and is perpendicular to the line y = 3x + 2.

Find the equation of the line that passes through the point (4, -1) and is perpendicular to the line y = 3x + 2.

Step 1: Identify the slope of the given line. The equation of the line is y = 3x + 2. The slope (m) is 3.
Step 2: Find the slope of the line that is perpendicular to the given line. The slope of a perpendicular line is the negative reciprocal of the original slope. So, the negative reciprocal of 3 is -1/3.
Step 3: Use the point-slope form of the equation of a line, which is y - y1 = m(x - x1). Here, (x1, y1) is the point (4, -1) and m is -1/3.
Step 4: Substitute the values into the point-slope form: y - (-1) = -1/3(x - 4). This simplifies to y + 1 = -1/3(x - 4).
Step 5: Distribute -1/3 on the right side: y + 1 = -1/3x + 4/3.
Step 6: Isolate y by subtracting 1 from both sides: y = -1/3x + 4/3 - 3/3, which simplifies to y = -1/3x + 1/3.
Step 7: The final equation of the line is y = -1/3x + 1/3.

Slope of a Line – Understanding how to find the slope from the equation of a line and how to determine the slope of a perpendicular line.
Point-Slope Form – Using the point-slope form of a linear equation to find the equation of a line given a point and a slope.