Problem 8 - Precalculus with Limits

Question

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

Step-by-Step Solution

Answer

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

Step 1: 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.

Step 2: 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.

Step 3: Match Equation (c)

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

Step 4: 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.

Step 5: 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.

Key Concepts

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.