Problem 35 - Precalculus Essentials

Question

If you know a point on a line and you know the equation of a line perpendicular to this line, explain how to write the line's equation.

Step-by-Step Solution

Answer

To find the equation of a line given a point on the line and the equation of a perpendicular line: (1) Identify the slope of the given line, (2) find the negative reciprocal of this slope to get the slope of the line we need, and (3) use the known point and the slope in the point-slope form of the line equation to get the required line's equation.

Step 1: Identify the Slope of the Perpendicular Line

First step is identifying the slope (m) of the given perpendicular line. If the equation is in the slope-intercept form (y = mx + b), the coefficient of x is the slope.

Step 2: Calculate the Slope of the Line

Since the lines are perpendicular, the slope of the line we want to find is the negative reciprocal of the slope of the given perpendicular line. In other words, if the slope of the perpendicular line is \( a \), then the slope of the line will be \( -1/a \).

Step 3: Write the Equation of the Line

Now we know the slope of the line and a point on it. Let's suppose that point is \((x_1, y_1)\) and the slope we calculated in Step 2 is \( b \). We can now use the point-slope form of a line equation, which is \( y - y_1 = b(x - x_1) \). Substitute \( x_1, y_1 \) and \( b \) into this equation to obtain the equation of the line.