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Chapter 2: Problem 12

Write an equation in slope-intercept form for the line that satisfies each set of conditions. perpendicular to \(y=\frac{1}{2} x+6,\) passes through \((-5,7)\)

Short Answer

Expert verified
The equation is \(y = -2x - 3\).

Step by step solution

01

Identify Slope of Given Line

The given line is in the form \(y = mx + b\), where \(m\) is the slope. For the line \(y = \frac{1}{2}x + 6\), the slope \(m\) is \(\frac{1}{2}\).
02

Determine Slope of Perpendicular Line

The slope of a line that is perpendicular to another line is the negative reciprocal of the original line's slope. Thus, the slope of the perpendicular line is \(-2\), since the negative reciprocal of \(\frac{1}{2}\) is \(-2\).
03

Use Point-Slope Form to Write the Equation

We use the point-slope form of a line equation to substitute the known point \((-5, 7)\) and the slope \(-2\). The formula is \(y - y_1 = m(x - x_1)\), where \(m\) is the slope and \((x_1, y_1)\) is the point. Substitute to get \(y - 7 = -2(x + 5)\).
04

Simplify to Slope-Intercept Form

Expand the equation \(y - 7 = -2(x + 5)\) to get it into slope-intercept form \(y = mx + b\). Distribute the \(-2\) to get \(y - 7 = -2x - 10\), and then add \(7\) to both sides to get \(y = -2x - 3\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Perpendicular Lines
Perpendicular lines are lines that intersect at a right angle, which is 90 degrees. When we are given a line, the perpendicular line to it has a special relationship with its slope.

- The slope of a given line is a measure of its steepness.- For two lines to be perpendicular, the product of their slopes must equal -1.

For example, if Line 1 has a slope of \( \frac{1}{2} \), the perpendicular slope will be the negative reciprocal, which is \(-2\). This relationship ensures that the two lines will always intersect at a right angle.
Slope
The slope of a line is a key concept in algebra and geometry. It quantifies the steepness and direction of a line.

- Slope is often represented by the letter \( m \).- The formula to calculate slope from two points, \((x_1, y_1)\) and \((x_2, y_2)\), is \( m = \frac{y_2 - y_1}{x_2 - x_1} \).

In this particular problem, we were originally provided a line in slope-intercept form, \( y = \frac{1}{2}x + 6 \), where the slope \( m = \frac{1}{2} \). When dealing with perpendicular lines, you need to find the negative reciprocal of the given slope for your new line.
Point-Slope Form
The point-slope form is used to find the equation of a line when you know a point on the line and its slope.

- The formula is \( y - y_1 = m(x - x_1) \).- Here \( m \) is the slope, and \( (x_1, y_1) \) are the coordinates of a known point on the line.

Using this formula makes it easy to start an equation with a specified slope and point. In our case, with a slope of \(-2\) and the point \((-5, 7)\), it translates to the equation: \( y - 7 = -2(x + 5) \). This intermediate step prepares the equation for conversion to a more familiar form.
Equation of a Line
An equation of a line is a mathematical expression that represents all the points along that line. There are several forms, but the slope-intercept form is one of the most common.

- The slope-intercept form is \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.- The y-intercept is the point where the line crosses the y-axis.

In the exercise, we transformed the point-slope equation \( y - 7 = -2(x + 5) \) into the slope-intercept form \( y = -2x - 3 \) by distributing and simplifying. This makes it easy to see both the slope \(-2\) and the y-intercept \(-3\) directly from the equation, which helps graph the line and understand its properties quickly.

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