Problem 29 - Elementary Algebra

Question

Find the equation of the line with the given slope and \(y\) intercept. Leave your answers in slope-intercept form. (Objective 1a) \(m=-1\) and \(b=\frac{5}{2}\)

Step-by-Step Solution

Answer

The equation is \( y = -x + \frac{5}{2} \).

Step 1: Identify the Formula

The equation of a line in slope-intercept form is given by the formula \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.

Step 2: Substitute the Given Values

Substitute the given slope \( m = -1 \) and y-intercept \( b = \frac{5}{2} \) into the slope-intercept formula. This gives us: \( y = -1 \cdot x + \frac{5}{2} \).

Step 3: Simplify the Equation

Simplify the expression by multiplying \( -1 \) by \( x \), which results in \( y = -x + \frac{5}{2} \). This is the equation of the line in slope-intercept form.

Key Concepts

Linear Equations

A linear equation is an equation that represents a straight line when graphed on a coordinate plane. It typically takes the form \( ax + by = c \), where \( a \), \( b \), and \( c \) are constants. However, one of the most common and useful forms of a linear equation for graphing is the slope-intercept form.
The slope-intercept form is expressed as \( y = mx + b \), where \( m \) is the slope of the line, and \( b \) is the y-intercept.
This format allows us to easily identify important features of the line, like its steepness (slope) and where it crosses the y-axis (y-intercept). These aspects are crucial when graphing the equation or understanding the line's behavior.

• The "slope" \( m \) determines how slanted the line is.
• The "y-intercept" \( b \) tells us the starting point of the line on the y-axis.

Linear equations are foundational in algebra, as they form the building blocks for more complex equations. They are used in various applications, including physics, economics, and everyday problem-solving scenarios.

Slope

The slope of a line is a measure of its steepness and direction. In the slope-intercept form \( y = mx + b \), \( m \) represents the slope.
Slope is calculated as "rise over run," meaning the change in the vertical direction (rise) over the change in the horizontal direction (run). It's a ratio given by \((y_2 - y_1) / (x_2 - x_1)\) for two points \((x_1, y_1)\) and \((x_2, y_2)\) on the line.
A positive slope indicates that the line ascends from left to right, while a negative slope means it descends. In our example where \( m = -1 \):

• The line goes down one unit vertically for every unit it goes to the right horizontally.
• This negative slope signifies a downward trend.

When graphed, the slope shows how much \( y \) changes with each unit increase in \( x \). Understanding slope is crucial for interpreting and predicting patterns in data and solving real-world challenges.

Y-Intercept

The y-intercept of a line refers to the point where the line crosses the y-axis. In the slope-intercept form \( y = mx + b \), \( b \) is the y-intercept.
Since this point is where the line meets the y-axis, it has coordinates \( (0, b) \). For lines in the slope-intercept form, the y-intercept provides a direct insight into the starting height of the line when \( x = 0 \).
In our specific equation, the y-intercept is \( b = \frac{5}{2} \), which means:

• The line crosses the y-axis at the point \( (0, \frac{5}{2}) \).

Understanding the y-intercept helps us quickly graph the line and comprehend its overall position relative to the y-axis. This concept is integral in graphing and analyzing linear relationships as it often represents the initial value of a scenario or experiment.