Problem 13 - Algebra 1: Concepts and Skills

Question

Write in slope-intercept form the equation of the line described below. $$ m=3, b=2 $$

Step-by-Step Solution

Answer

The equation of the line in slope-intercept form is \(y = 3x + 2\).

Step 1: Identify the given slope and y-intercept

From the given instruction, the slope 'm' is identified as 3 and the y-intercept 'b' is identified as 2.

Step 2: Substitute the slope and intercept into the slope-intercept form

Substitute 'm' with 3 and 'b' with 2 in the formula \(y = mx + b\). This gives the equation \(y = 3x + 2\)

Key Concepts

Equation of a Line

An equation of a line describes a straight path on a graph. It shows the relationship between the variable "x" and the variable "y," where every pair

• "(x, y)" satisfies the line's equation.
• Each point on the line is a solution to the equation.
• The line can be represented in different forms, such as the standard form, point-slope form, and slope-intercept form.

The slope-intercept form, which is given by the formula \(y = mx + b\), is one of the most familiar and straightforward ways to represent a line. Here, "\(m\)" is the slope determining the line's steepness, and "\(b\)" is the y-intercept, where the line crosses the y-axis. This form is particularly useful because it directly reveals both the slope and y-intercept, which are crucial for drawing the line and understanding its orientation on a graph. Understanding how to write and interpret this form is fundamental in algebra and is the simplest way to describe linear relationships between two variables.

Slope

The slope of a line is a measure of its steepness and direction. It is represented by the letter "\(m\)" in the slope-intercept form of an equation. Mathematically, the slope is defined as the ratio of the "rise" (vertical change) to the "run" (horizontal change) between two points on the line.

The formula to determine the slope is:\[m = \frac{{\Delta y}}{{\Delta x}} = \frac{{y_2 - y_1}}{{x_2 - x_1}}\]This means that for every unit increase in "x," the value of "y" changes by the amount of the slope "m."

• A positive slope means the line ascends from left to right.
• A negative slope indicates the line descends from left to right.
• When the slope is zero, the line is horizontal.
• An undefined slope corresponds to a vertical line.

In the example exercise, the slope \(m\) is given as 3. This tells us that for every unit increase in "x," the value of "y" will increase by 3 units, implying a steep upward line.

Y-Intercept

The y-intercept is the point where a line crosses the y-axis. In the slope-intercept form, the y-intercept is denoted by the letter "\(b\)."

It represents the value of "y" when "x" is zero, serving as the starting point of the line when graphing.

• If \(b\) is positive, the line crosses above the origin.
• If \(b\) is negative, the crossing occurs below the origin.
• A y-intercept of zero means the line passes through the origin itself.

In our exercise, the y-intercept is 2. This means the line will intersect the y-axis at the point (0, 2). Therefore, without any increase or decrease along the "x" axis, the line starts at 2 on the "y" axis. Understanding the y-intercept helps plot the initial point when drawing the graph of the equation \(y = mx + b\). Knowing the starting point and the slope allows you to graph the entire line accurately.