Problem 10 - Algebra 1: Concepts and Skills

Question

Write the equation of the line passing through the point and perpendicular to the given line. $$ (6,0), y=-2 x+7 $$

Step-by-Step Solution

Answer

The equation of the line passing through the point (6,0) and perpendicular to the given line \(y=-2x+7\) is \(y=0.5x-3\).

Step 1: Find the Slope of the Given Line

First, identify the slope of the given line. The equation of the given line is in slope-intercept form \(y=mx+c\), where \(m\) is the slope. Here, the slope of given line is -2.

Step 2: Determine the Slope of the Perpendicular Line

The slope of a line perpendicular to the given line is the negative reciprocal of the given line's slope. Therefore, the slope of the perpendicular line is \(-1/-2=0.5\).

Step 3: Use Point-Slope Form to Find the Equation of the Line

The point-slope form of a line equation is \(y-y1=m(x-x1)\). Here, \(m\) is the slope of the line, and \((x1,y1)\) is a point on the line. We know the slope (\(m = 0.5\)) and a point \((x1,y1) = (6,0)\) on the line. By substituting these values, we obtain \(y-0 = 0.5(x-6)\), which simplifies to \(y=0.5x-3\).

Key Concepts

Slope-Intercept Form

Understanding the slope-intercept form is crucial when dealing with linear equations. It's the classic 'y=mx+b', where 'm' represents the slope, and 'b' is the y-intercept. This straightforward format allows us to quickly interpret the graph of a line.

In study situations, students often encounter linear equations presented in this form because it shows how a line moves as 'x' changes. The slope 'm' indicates the line's steepness and direction, rising or falling as you move along the x-axis. The y-intercept 'b' tells us where the line crosses the y-axis, providing a starting point for drawing the line or understanding where it exists in relation to the origin.

With the exercise in question, we immediately spot the slope 'm' as -2, and the y-intercept 'b' as +7. Therefore, each time 'x' increases by one, 'y' decreases by two, plotting a line descending from left to right.

Point-Slope Form

When you know a single point on a line and its slope, the point-slope form equation, 'y - y1 = m(x - x1)', is the way to go. This form highlights the relationship between the identified point '(x1, y1)' and any other point '(x, y)' on the line. The 'm' still stands for the slope of the line.

Moving from theory to practice, this form is instrumental in constructing the equation of the line in our exercise. After finding the slope of the perpendicular line to be 0.5, point-slope form uses that slope and the point (6,0). Simply plug the values into the formula to get 'y - y1 = 0.5(x - x1)'. By substituting (6,0) for '(x1, y1)', the point-slope equation becomes 'y - 0 = 0.5(x - 6)', which simplifies to the desired equation 'y = 0.5x - 3'.

Negative Reciprocal Slope

The concept of negative reciprocal slopes is pivotal in determining the relationship between perpendicular lines. When two lines are perpendicular, their slopes have a distinct connection: they are negative reciprocals of each other. That means if one line has a slope of 'a', the perpendicular line will have a slope of '-1/a'.

Why is this important for solving our textbook problem? Once we've identified the original slope of -2, we apply the negative reciprocal rule to find the perpendicular slope: '-1/(-2)' which equals '0.5'. This fundamental step sets the stage for everything that follows. Students should, therefore, familiarize themselves with flipping and negating slopes, as mastering this pattern unlocks the ability to solve any perpendicular line equation.