Problem 22 - Elementary Algebra

Question

For the following problems, write the equation of the line using the given information in slope-intercept form. $$ m=1,(3,8) $$

Step-by-Step Solution

Answer

Answer: The equation of the line is $$y = x + 5$$.

Step 1: Identify the given information.

We have the following given information: Slope (m) = 1 Point on the line (x1, y1) = (3, 8)

Step 2: Apply the point-slope form

Using the point-slope form: y - y1 = m(x - x1) Substitute the given values, so we have the following equation: y - 8 = 1(x - 3)

Step 3: Convert equation into slope-intercept form

To convert the equation into slope-intercept form (y = mx + b), first, distribute the slope (m) to both x and -x1: y - 8 = 1 * x - 1 * 3 y - 8 = x - 3 Now, add 8 to both sides of the equation to isolate y on one side: y = x - 3 + 8 Finally, simplify the equation to get the line in slope-intercept form: y = x + 5 The equation of the line in slope-intercept form is: $$y = x + 5$$.

Key Concepts

Point-Slope Form

The point-slope form of a linear equation is an incredibly useful formula that makes it easy to create the equation of a line as long as you have two key pieces of information: the slope of the line and one point through which the line passes.
The general formula for point-slope form is:

• \( y - y_1 = m(x - x_1) \)

Here, \( m \) stands for the slope, \( (x_1, y_1) \) is the given point on the line, and \( y \) and \( x \) are variables representing any point on the line.
This format emphasizes the line's change in y in relation to its change in x, allowing us to easily substitute in the known point and the slope to find specific equations for specific lines.

Linear Equations

Linear equations are equations that form straight lines when graphed on a coordinate plane. These equations are represented by the general form \( y = mx + b \), where the graph of the equation is a straight line.
Here:

• \( m \) is the slope of the line.
• \( b \) is the y-intercept, the point where the line crosses the y-axis.

The slope-intercept form is particularly user-friendly since it allows for quick identification of both the slope and the y-intercept, providing a clear picture of the line's behavior.
Understanding how to rewrite equations in this form is crucial because it forms the basis for graphing lines and analyzing their intersections, which is a common task in algebra and higher-level math.

Slope of a Line

The slope of a line, denoted as \( m \), is a key concept in understanding how linear equations work. The slope indicates how steep a line is and in which direction it is heading on a graph.
Mathematically, the slope is calculated by the formula:

• \( m = \frac{\Delta y}{\Delta x} \)

This formula means that the slope is the ratio of the change in the vertical direction (\( \Delta y \)) to the change in the horizontal direction (\( \Delta x \)) between two distinct points on a line.
A positive slope means that the line inclines upward as it moves from left to right. Conversely, a negative slope indicates the line declines. When the slope is zero, the line is perfectly horizontal; this signifies that there is no vertical change.
Understanding the slope is vital when creating or interpreting the equations of lines, as it determines the line's steepness and direction.