Problem 15 - Elementary Algebra

Question

For the following problems, write the equation of the line using the given information in slope-intercept form. $$ m=2, y \text { -intercept }(0,5) $$

Step-by-Step Solution

Answer

Question: Write the equation of a line with a slope of 2 and a y-intercept of (0, 5) in slope-intercept form. Answer: y = 2x + 5

Step 1: Identify the slope and y-intercept

We are given the slope (m) as 2 and the y-intercept (b) as (0, 5). We know that the y-intercept is the point at which the line crosses the y-axis. In this case, the point is (0, 5), so b = 5.

Step 2: Write the equation of the line in slope-intercept form

Given the values of m and b from Step 1, substitute them in the slope-intercept form, which is y = mx + b. $$ y = 2x + 5 $$ This is the equation of the line in slope-intercept form.

Key Concepts

Equation of a Line

In mathematics, an equation of a line is used to describe a straight line on a graph. This equation is crucial as it provides a way to express any line in a coordinate plane using algebraic expressions. The most common form of a linear equation is the **slope-intercept form**:

• The equation is written as \( y = mx + b \), where \( y \) and \( x \) represent the coordinates on the y-axis and x-axis, respectively.
• Here, \( m \) is the slope of the line, and \( b \) is the y-intercept.

Essentially, this formula allows one to easily plot the line on a graph and understand how it behaves in relation to the axes. The **slope-intercept form** is particularly popular because of its simplicity and the ease with which it describes the properties of the line in a glance.

Slope

The slope of a line, often represented by \( m \), is a measure of the line's steepness. It's calculated as the ratio of the rise (the change in y) over the run (the change in x). Simply put:

• Slope \( m \) = \( \frac{\Delta y}{\Delta x} \)

Here, \( \Delta y \) is the difference in the y-values (vertical change) and \( \Delta x \) is the difference in the x-values (horizontal change). The slope tells us if the line is:

• Positive (slopes upward from left to right)
• Negative (slopes downward from left to right)
• Zero (horizontal line)
• Undefined (vertical line)

Understanding the slope helps in predicting how a line moves across the plane. For the slope \( m = 2 \), it means for every unit you move to the right on the x-axis, the line goes up 2 units on the y-axis.

Y-Intercept

The y-intercept of a line is a point where the line crosses the y-axis. This is a significant part of a line equation, represented as \( b \) in the slope-intercept form, \( y = mx + b \). At the y-intercept, the value of \( x \) is zero.

• This makes the y-intercept easy to identify in a graph as it will always be at the point \( (0, b) \).

In our case, the y-intercept is \( 5 \), which means the line will cross the y-axis at the point \( (0, 5) \). Knowing the y-intercept allows you to draw the line easily without needing multiple points as it provides a fixed point through which the line will pass.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is the study of geometry using a coordinate system. This branch of geometry is vital because it bridges algebra and geometry through graphs and equations.

• With coordinate geometry, we can classify shapes, explore their properties, and understand relationships between different geometric figures.
• It typically involves plotting points, lines, and shapes on a graph to solve problems involving distances, midpoints, and slopes.

In problems such as finding the equation of a line, coordinate geometry is employed to interpret the position and trajectory of the line on the Cartesian plane. This tool helps in solving real-world problems by offering a visual representation that makes understanding geometric concepts more intuitive.