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Chapter 1: Problem 8

Match each equation of a line with its form. (a) \( Ax + By + C = 0 \) (b) \( x = a \) (c) \( y = b \) (d) \( y = mx + b \) (e) \( y - y_1 = m(x-x_1) \) (i) Vertical line (ii) Slope-intercept form (iii) General form (iv) Point-slope form (v) Horizontal line

Short Answer

Expert verified
(a) matches with (iii), (b) matches with (i), (c) matches with (v), (d) matches with (ii), (e) matches with (iv)

Step by step solution

01

Match Equation (a)

(a) \(Ax + By + C = 0\) - This is the general form of a linear equation and matches with (iii) General form. Here A, B and C are constants.
02

Match Equation (b)

(b) \(x = a\) - This is an equation of a vertical line where x is a constant. It matches with (i) Vertical line.
03

Match Equation (c)

(c) \(y = b\) - This equation describes a horizontal line where y is a constant. It corresponds to (v) Horizontal line.
04

Match Equation (d)

(d) \(y = mx + b\) - This is the slope-intercept form of the linear equation where m is the slope and b is the y-intercept. It matches with (ii) Slope-intercept form.
05

Match Equation (e)

(e) \(y - y_1 = m(x-x_1)\) - This is the point-slope form of the linear equation where \(y_1\) and \(x_1\) are coordinates of a point on the line and m is the slope. It corresponds to (iv) Point-slope form.

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Key Concepts

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

General Form of a Linear Equation
The general form of a linear equation is depicted as Ax + By + C = 0, where A, B, and C are real-number constants, and x and y are variables. This form is useful for analytically describing lines on a coordinate plane, and it enables us to work with lines even when the slope may not be readily apparent.

For example, if we have the equation 3x + 2y - 6 = 0, it is in the general form. We can manipulate this form to find the slope-intercept or point-slope versions, which are often easier to visualize. To do that, we solve the equation for y.
Slope-Intercept Form
The slope-intercept form of a linear equation is one of the most common ways to represent a line. It is written as y = mx + b, where m represents the slope, and b represents the y-intercept - the point where the line crosses the y-axis. This form is particularly useful for quickly sketching a graph of the line. The slope indicates the steepness and the direction of the line.

For instance, the equation y = 2x + 1 tells us that for every one unit increase in x, y increases by two units, and the line crosses the y-axis at (0, 1).
Point-Slope Form
The point-slope form is expressed as y - y_1 = m(x - x_1), where m is the slope and (x_1, y_1) represents the coordinates of a specific point on the line. This form is useful for when you have the slope of a line and one point through which the line passes.

For example, if a line has a slope of 3 and passes through the point (2, -1), the point-slope form would be y + 1 = 3(x - 2). This can then be transformed into either the slope-intercept or general form if necessary.
Vertical Line
The equation of a vertical line is unique in that it cannot be expressed in slope-intercept or point-slope form since vertical lines have an undefined slope. It is always in the form of x = a, where a is the x-coordinate of all points on the line. Essentially, this means every point on the line has the same x-coordinate.

For instance, the line x = 3 means that no matter what the value of y is, x will always be 3. This line is a straight vertical line crossing the x-axis at (3, 0).
Horizontal Line
A horizontal line's equation is as straightforward as that of a vertical line and is shown as y = b, where b is the y-coordinate of all points on the line. Similar to a vertical line, horizontal lines have a slope of zero.

The line y = -2, for instance, means that the line is flat, parallel to the x-axis, and crosses the y-axis at (0, -2). Regardless of the x-coordinate, the y-coordinate is consistently -2 for any point on this line.

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