Problem 15 - Algebra 1: Concepts and Skills

Question

Write in slope-intercept form the equation of the line described below. $$ m=0, b=6 $$

Step-by-Step Solution

Answer

The line's equation in slope-intercept form is \(y=6\).

Step 1: Identify the values

From the exercise, the slope (m) is given as 0 and the y-intercept (b) is given as 6.

Step 2: Substitution into the slope-intercept equation

Substitute your 'm' and 'b' values into the general slope-intercept equation which is \(y=mx+b\). So you can replace 'm' with 0 and 'b' with 6 which gives us \(y= 0x+6.\)

Step 3: Simplification

The term \(0x\) simplifies to 0, since any number multiplied by 0 equals 0. Therefore, the simplified equation of the line in slope-intercept form is \(y=6\).

Key Concepts

Linear Equations

A linear equation represents a straight line on a Cartesian plane and is one of the fundamental concepts in algebra. It is essential for students to recognize that any linear equation in two variables can be written in the form of ax + by = c, where a, b, and c are real numbers and x and y are the variables. One popular way to express a linear equation is in slope-intercept form, which is written as y = mx + b. Here, m represents the slope of the line, which describes its steepness, and b signifies the y-intercept, the point where the line crosses the y-axis.

In our exercise, we work directly with the slope-intercept form to define the line with a slope of zero and a y-intercept of six. To ensure clear understanding, it's helpful to know that a zero slope indicates a horizontal line. The concept becomes complete when students can visualize this through graphing the equation and observing the horizontal line crossing the y-axis at the point (0,6).

Slope of a Line

The slope of a line represents the rate of change between the y-coordinates and the x-coordinates of any two points on the line. Mathematically, it is the rise over run, or the vertical change divided by the horizontal change. In the slope-intercept form \(y = mx + b\), the slope is denoted by m.

For instance, if you have two points on a line, point (x1, y1) and point (x2, y2), the slope is calculated as \( m = \frac{{y2 - y1}}{{x2 - x1}} \). If the slope is positive, the line rises as it moves from left to right. Conversely, if the slope is negative, the line falls. A slope of zero, as in our example equation \(y = 0x + 6\), indicates a horizontal line, which has no vertical change regardless of the horizontal movement.

Y-intercept

The y-intercept is where a line intersects the y-axis of a graph. In an equation of the form \(y = mx + b\), the y-intercept is represented by the constant term b. It is the exact point where the x-coordinate is zero. This specific value of y when x equals zero can be thought of as the starting point of the line on the Cartesian plane.

In practical terms, understanding the y-intercept can help students grasp how a particular situation begins or the initial value in a real-world context. For our exercise, where the equation is \(y = 6\), the y-intercept is (0, 6), suggesting that the line starts at six units above the origin on the y-axis and remains constant since the slope is zero.