Problem 22 - Algebra 1

Question

Write an equation of the line that passes through the point and has the given slope. Write the equation in slope-intercept form. $$(3,4), m=0$$

Step-by-Step Solution

Answer

The equation of the line in slope-intercept form that passes through (3,4) and has a slope of 0 is \(y=4\).

Step 1: Identify the given point and slope

The provided point through which the line passes is (3,4) and the slope of the line given as m is 0.

Step 2: Use slope-intercept form

The slope-intercept form of a linear equation is represented as \(y=mx+b\) where m is the slope of the line and b is the y-intercept (the value of y when x equals 0). Substituting the given slope (m) into the formula gives us \(y=0x+b\), which further simplifies to \(y=b\).

Step 3: Find the y-intercept

The y-intercept (b) is the value of y at the point where the line crosses the y-axis. But since this line has a slope of 0, it's a horizontal line and doesn't intersect the y-axis at a point other than the y-coordinate of the given point. Therefore, the y-intercept is equal to the y-coordinate of the given point, which in this case is 4.

Step 4: Write final equation

Substitute the y-intercept (b) into the equation from Step 2 to obtain the coefficient of y in the equation. This yields the final equation: \(y=4\)

Key Concepts

Understanding Slope

Slope is an essential concept in understanding linear equations. It essentially describes how "steep" a line is. In a more mathematical sense, slope is a number that represents the ratio of the "rise" (the vertical change) to the "run" (the horizontal change) between two points on a line.
The formula for finding the slope, often denoted by the letter \(m\), is:

• \( m = \frac{\text{rise}}{\text{run}} = \frac{y_2-y_1}{x_2-x_1} \)

This formula calculates the steepness by showing how much \(y\) changes as you move horizontally from \(x_1\) to \(x_2\).
In the example given, the slope \(m\) is \(0\). What does this mean? If the slope \(m\) is 0, then the line is perfectly horizontal. That means there is no change in \(y\) (no rise), regardless of how much \(x\) changes. The lack of vertical change results in a flat or horizontal line across the graph.

Slope-Intercept Form Demystified

The slope-intercept form of a linear equation is one of the most common ways to express a straight line. Written as:

• \(y = mx + b\)

This format is very useful because it directly shows the slope \(m\) and the y-intercept \(b\) of the line.
Why is this form so practical? The slope-intercept form lets you quickly identify key characteristics of the line just by looking at the equation. With \(y = mx + b\):

• \(m\) is the slope: It indicates the direction or angle of the line as it moves away from the y-axis.
• \(b\) is the y-intercept: It tells you the exact point where the line will cross the y-axis.

In our exercise, the equation simplifies to just \(y = b\) because the slope \(m\) equals 0. This highlights that the line remains constant across all \(x\) values, reinforcing the idea of a horizontal line.

What is a Y-Intercept?

The y-intercept is a crucial concept because it marks the point where a line crosses the y-axis. Essentially, it tells you the value of \(y\) when \(x\) is equal to 0.
In slope-intercept form, \(y = mx + b\), the \(b\) represents the y-intercept. For any line, this is where the graph will intersect or "meet" the y-axis. This point can be easily identified when \(x\) is 0 because it simplifies the equation to \(y = b\).
In the given problem, since the line is horizontal with a slope \(m\) of 0, the line doesn’t rise or fall, meaning it runs parallel to the x-axis. As such, the y-intercept will be the actual y-coordinate of the given point. For the point (3, 4), this makes our y-intercept \(b = 4\).This results in a simple equation of \(y = 4\), clearly showing that all points on the line share the same y-coordinate of 4, confirming the horizontal nature of the line.