Find the equation of the line that passes through (2, 3) and is perpendicular to

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

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

Step 1: Identify the slope of the given line. The equation of the line is y = 4x - 1. The slope (m) is 4.
Step 2: Find the slope of the line that is perpendicular to the given line. The perpendicular slope is the negative reciprocal of the original slope. So, the perpendicular slope is -1/4.
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 (2, 3) and m is -1/4.
Step 4: Substitute the values into the point-slope form: y - 3 = -1/4(x - 2).
Step 5: Simplify the equation. First, distribute -1/4: y - 3 = -1/4x + 1/2.
Step 6: Add 3 to both sides to isolate y: y = -1/4x + 1/2 + 3.
Step 7: Convert 3 to a fraction with a common denominator: 3 = 6/2. Now, y = -1/4x + 1/2 + 6/2.
Step 8: Combine the fractions: y = -1/4x + 7/2.
Step 9: The final equation of the line is y = -1/4x + 7/2.

Slope of a Line – Understanding how to determine the slope from the equation of a line and how to find 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.
Equation of a Line – Converting from point-slope form to slope-intercept form to express the equation of the line.