Problem 60 - Elementary Algebra

Question

Write the equation of the line using the given information. Write the equation in slope-intercept form. Slope \(=1, \quad y\) -intercept \(=8\)

Step-by-Step Solution

Answer

Question: Given the slope \(m = 1\) and a y-intercept \(b = 8\), write the equation of the line in slope-intercept form. Answer: The equation of the line is \(y = x + 8\).

Step 1: Recall the slope-intercept form of a linear equation

The slope-intercept form of a linear equation is: \(y = mx + b\), where \(m\) represents the slope and \(b\) represents the y-intercept of the line.

Step 2: Identify the slope and y-intercept

We are given the slope, \(m=1\), and the y-intercept, \(b=8\).

Step 3: Substitute the given values into the slope-intercept form

Now, we'll substitute the given values of the slope and y-intercept into the slope-intercept form. \(y = (1)x + (8)\)

Step 4: Write the final equation

The equation of the line in slope-intercept form is: \(y = x + 8\)

Key Concepts

Linear Equations

A linear equation is a fundamental concept in algebra. It represents a straight line when graphed on a coordinate plane. The general form of a linear equation is

• \( ax + by = c \)

where \(a\), \(b\), and \(c\) are constants.
Linear equations can be manipulated into different forms, such as the standard form, point-slope form, and slope-intercept form, each providing unique insights into the geometric properties of the line they represent.
These equations are called 'linear' because they graph as straight lines. This comes from the fact that they represent the relationship between two variables with a constant rate of change, or slope. In many real-world applications, linear equations are used to model relationships that assume this constant rate of change.

Slopes

The slope of a line measures its steepness and direction. In mathematics, slope is described using the letter \(m\). It is defined as

• the ratio of the change in y (rise) over the change in x (run).

This can be expressed mathematically as: \[m = \frac{\Delta y}{\Delta x}\]where "\(\Delta y\)" denotes the change in the vertical direction, and "\(\Delta x\)" represents the change in the horizontal direction.
A positive slope means the line ascends from left to right, whereas a negative slope descends. If a slope is zero, the line is horizontal, indicating no change in \(y\) as \(x\) changes. Conversely, an undefined slope implies a vertical line, where \(x\) doesn't change despite a change in \(y\). Understanding slope helps predict values and analyze the trend of data points in a graph.

Y-Intercepts

The y-intercept is a significant feature of a line graphed in a coordinate plane. It indicates where the line crosses the y-axis. In the slope-intercept form of a line, which is

• \( y = mx + b \)

"\(b\)" represents the y-intercept.
This value can be identified easily as it is the constant term when the equation is arranged in slope-intercept form.
The y-intercept tells us the value of \(y\) when \(x\) is zero. It provides a starting point for graphing the line. For instance, in the equation \(y = x + 8\), the y-intercept is 8, showing that the line crosses the y-axis at (0, 8). Identifying the y-intercept can be crucial in solving real-world problems, helping to determine initial conditions or starting values in various contexts.