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Chapter 2: Problem 13

Find an equation of the line that passes through the point and has the indicated slope \(m\). $$ (-3,2) ; m=0 $$

Short Answer

Expert verified
The equation of the line that passes through the point \((-3,2)\) and has the indicated slope \(m=0\) is \(y=2\).

Step by step solution

01

Identify the given point and slope

The point given is \((-3, 2)\). The slope given is \(m = 0\). We will use these values in the point-slope form of the linear equation.
02

Substitute the point and slope into the point-slope form

The point-slope form is given by $$ y - y_1 = m(x - x_1) $$ Substitute \((-3, 2)\) for \((x_1, y_1)\) and \(0\) for \(m\): $$ y - 2 = 0(x - (-3)) $$
03

Simplify the equation

Simplify the right side of the equation: $$ y - 2 = 0(x + 3) $$ Since \(0\) multiplied by any value is \(0\), we can further simplify the equation: $$ y - 2 = 0 $$
04

Solve for y

Add 2 to both sides of the equation to solve for y: $$ y = 2 $$
05

Write the final equation

We have found the equation of the line with a slope of 0 and passing through the point \((-3, 2)\): $$ y = 2 $$

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Key Concepts

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

Point-Slope Form
The point-slope form of a linear equation is incredibly useful when we know a single point on a line and its slope. This form is expressed as \( y - y_1 = m(x - x_1) \), where \( m \) is the slope of the line and \( (x_1, y_1) \) is the known point.

By plugging in the specific numbers for the slope \( m \) and the coordinates of the point \( x_1 \) and \( y_1 \) into this equation, we can easily write the equation for the line. This formula provides a direct method to create the base of our linear equation before further simplification.
Slope of a Line
The slope \( m \) is a measure of the steepness or the inclination of a line. It is calculated by comparing the difference in the y-coordinates (vertical change) to the difference in the x-coordinates (horizontal change) between two distinct points on the line.

The formula to find the slope when given two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is \( m = \frac{(y_2 - y_1)}{(x_2 - x_1)} \). In our exercise, the slope is given as zero, \( m = 0 \), indicating that the line is perfectly horizontal which we'll discuss in the next section.
Horizontal Lines
Horizontal lines are a special case in the world of linear equations as they have a slope of zero. They run left to right and are parallel to the x-axis.

Because the slope is zero, the equation simplifies dramatically. In our example, no matter what x-value you choose, the y-value will always be the same, represented by the constant term in the simplified equation \( y = 2 \). This reflects the fact that all the points on a horizontal line share the same y-coordinate.
Equation of a Line
An equation of a line provides a relationship between the x and y coordinates of any point on that line. Linear equations can be written in various forms including point-slope form, slope-intercept form, and standard form.

In cases like our exercise where the line is horizontal, the equation of the line is particularly simple, as it only features the y-coordinate. Thus, the final form of our linear equation, given the slope of zero and a passing through point \( (-3, 2) \) in the point-slope form, simplifies to \( y = 2 \), indicating that our line is a horizontal line crossing the y-axis at \( y = 2 \).

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