Problem 43 - College Algebra with Modeling and Visualization

Question

Find the slope-intercept form for the line satisfying the conditions. Perpendicular to \(x+y=4,\) passing through \((15,-5)\)

Step-by-Step Solution

Answer

The line's equation is \(y = x - 20.\)

Step 1: Write the equation in slope-intercept form

To find the slope of the line perpendicular to the given line, we first need to express the equation \(x+y=4\) in the slope-intercept form \(y=mx+b\). This is done by solving for \(y\): \[y = -x + 4.\] Thus, the slope \(m\) of the given line is \(-1\).

Step 2: Find the slope of the perpendicular line

The slope of a line perpendicular to another is the negative reciprocal of the original line's slope. Thus, the slope for our desired perpendicular line is \(1\). This is calculated by taking the negative reciprocal of \(-1\): \(m = -\frac{1}{-1} = 1\).

Step 3: Use the point-slope form of the equation

With the slope \(m = 1\) and a point \((15, -5)\) through which the line passes, use the point-slope equation: \[y - y_1 = m(x - x_1).\] Substituting the values: \[y + 5 = 1(x - 15).\]

Step 4: Convert to slope-intercept form

Distribute and simplify the equation: \[y + 5 = x - 15.\] Subtract 5 from both sides to get \(y\) alone: \[y = x - 20.\] This is the slope-intercept form of the desired line.

Key Concepts

Perpendicular Lines

Understanding perpendicular lines is crucial when working with linear equations. Perpendicular lines are two lines that intersect at a right angle (90 degrees). In terms of their slopes, two lines are perpendicular if the product of their slopes is -1. For instance, if one line has a slope of -1, a line perpendicular to it will have a slope that is the negative reciprocal, which in this case is 1. When you visualize this on a graph, perpendicular lines will appear as though they cross each other to form a 'T' shape. This concept is particularly important in geometry and can help determine angles and intersections in various mathematical problems.

Point-Slope Form

The point-slope form is a useful way to express the equation of a line when you know a point on the line and its slope. It is written as:

• \(y - y_1 = m(x - x_1)\)

where \(m\) is the slope, and \((x_1, y_1)\) is a known point on the line.

This form is particularly handy when needing to quickly develop the equation of a line based on limited information. For example, if you know that a line passes through the point (15, -5) with a slope of 1, you can plug those values into the point-slope formula to find the equation of the line. This method is efficient and helps simplify the process of finding a linear equation.

Negative Reciprocal

Calculating the negative reciprocal is essential when dealing with perpendicular lines. The negative reciprocal of a number is found by inverting the number and changing its sign. For example:

• The negative reciprocal of -1 is 1, because inverting -1 gives -1/1, and changing the sign results in 1.

In the context of slope, if you know one line's slope, you can find the slope of a perpendicular line by taking the negative reciprocal. This process ensures that the two lines will intersect at a right angle. Understanding negative reciprocals helps in correctly identifying relationships between lines in geometric figures and algebraic equations.

Linear Equations

Linear equations represent straight lines and can be written in multiple forms, including slope-intercept form and point-slope form.Slope-intercept form is given by:

• \( y = mx + b \)

where \(m\) represents the slope and \(b\) the y-intercept.

Linear equations describe relationships where a change in one variable is directly proportional to a change in another.

By manipulating and changing forms, you can derive various useful properties of lines, such as finding intersections, distances between points, and understanding geometric shapes.