Problem 62 - Introductory and Intermediate Algebra for College Students 4th

Question

Write an equation in slope-intercept form of the line satisfying the given conditions. The line passes through \((-2,6)\) and is perpendicular to the line whose equation is \(x=-4\)

Step-by-Step Solution

Answer

The slope-intercept form of the line is \(y = 6\).

Step 1: Identify the slope of the line

For a line perpendicular to a vertical line like x=-4, the slope is 0 because it is a horizontal line.

Step 2: Identify the y-intercept

The y-intercept is the y-coordinate of the point where the horizontal line intersects the y-axis. Because it is a horizontal line and it passes through the point (-2,6), the y-coordinate for all points including the y-intercept is 6.

Step 3: Write the equation in slope-intercept form

Substituting the slope (m=0) and y-intercept (b=6) into the slope-intercept form (y=mx+b), the equation of the line is y=0*x+6, which simplifies to y=6.

Key Concepts

Perpendicular Lines

Understanding perpendicular lines is pivotal in geometry and algebra, especially when dealing with coordinate systems. Perpendicular lines are two lines that intersect at a 90-degree angle. You can visually identify them by their 'T' shaped formation when they cross.

In the coordinate plane, the slopes of perpendicular lines have a particular relationship — they are negative reciprocals of each other. This means that if the slope of one line is 'm', then the slope of the line perpendicular to it will be '-1/m'. However, there is an exception to this rule: vertical and horizontal lines.

Exception for Vertical and Horizontal Lines

Vertical lines have an undefined slope because they rise infinitely without running horizontally. So, when a line is perpendicular to a vertical line, it must be horizontal, which means it has a slope of 0. Therefore, the slope of our line, which is perpendicular to the vertical line given by the equation x=-4, is 0. We don't have to calculate the negative reciprocal because vertical and horizontal lines are always perpendicular by definition.

Horizontal Line Equation

In coordinate geometry, a horizontal line is a straight line that extends from left to right or right to left and has a constant y-coordinate value for all points on the line. It's like the horizon you see when you look at the sea — perfectly flat and level.

Equation Form of Horizontal Lines

The equation for a horizontal line is very simple: it's always written as 'y = c' where 'c' is the constant y-coordinate of any point on the line. That's because the slope, 'm', of a horizontal line is 0, so when we use the slope-intercept form of a line (y=mx+b), the 'mx' part is always zero, leaving us with 'y = b'.

For our exercise, since the line is horizontal and passes through (-2,6), the y-coordinate (and thus the y-value in our equation) is constant at 6, leading us to the equation y=6. This tells us that no matter what value x takes, y will always be 6.

Y-Intercept

The y-intercept of a line is the point at which the line crosses the y-axis on a coordinate plane. It's like the starting point on a ladder that is leaning against a wall; the bottom of the ladder is the y-intercept.

Importance of the Y-Intercept in Equations

When you write the equation of a line in slope-intercept form, 'y=mx+b', 'b' represents the y-intercept. It's the value of y when x is 0. In our exercise, determining the y-intercept was straightforward since the line passes through the point (-2,6) and is horizontal. Since horizontal lines have the same y-coordinate for all points, the y-intercept is simply the y-coordinate of the given point, which is 6. So, in the slope-intercept equation y=mx+b, 'b' is 6, providing additional confirmation that the line's equation is y=6.

In summary, the y-intercept is a key component in the slope-intercept form and provides vital information for graphing the line on a coordinate plane.