Problem 33 - College Algebra 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 line equation, take the negative reciprocal of the slope from the given line to get the slope of your line, then use the point-slope form of equation, \(y - y_1 = m(x - x_1)\), and finally rewrite it in slope-intercept form if necessary.

Step 1: Identify the slope of the given line

The slope of the given line can be found by identifying the coefficient of \(x\) in the equation. If the given line equation is \(y = mx + c\), then the slope is \(m\).

Step 2: Find the slope of the line we need to write

Knowing that the slopes of two perpendicular lines are negative reciprocals of one another, find the slope of the line we need to write by taking the negative reciprocal of the slope from the given line. It means if the given line's slope is \(m\), our line's slope will be \(-1/m\).

Step 3: Use the point-slope form of a line

With the new slope and the given point \((x_1, y_1)\), the line's equation can be written using the point-slope form \(y - y_1 = m(x - x_1)\), where \(m\) is the slope from the previous step.

Step 4: Put the equation in slope-intercept form

If needed, rewrite the equation from step 3 into the slope-intercept form \(y = mx + c\). That would require some algebra manipulation and simplification.