Problem 30 - 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) \(m=-2\) and \(b=\frac{7}{3}\)

Step-by-Step Solution

Answer

The equation is \( y = -2x + \frac{7}{3} \).

Step 1: Understand the Slope-Intercept Form

The slope-intercept form of a linear equation provides a way to easily identify the slope and the y-intercept. It is given by the equation \( y = mx + b \), where \( m \) represents the slope and \( b \) is the y-intercept.

Step 2: Substitute the Given Slope

Replace \( m \) in the slope-intercept equation with the given slope value of \(-2\). Thus, the equation becomes \( y = -2x + b \).

Step 3: Substitute the Given Y-Intercept

Replace \( b \) in the equation \( y = -2x + b \) with the given value of \( \frac{7}{3} \). Now the equation becomes \( y = -2x + \frac{7}{3} \).

Step 4: Final Slope-Intercept Equation

The equation of the line that satisfies the given conditions is \( y = -2x + \frac{7}{3} \). It is now in slope-intercept form with a slope of \(-2\) and a y-intercept of \(\frac{7}{3}\).

Key Concepts

Linear Equations

Linear equations form the foundation of algebra and consist of terms that are either constants or the product of a constant and a single variable. They are called 'linear' because they graph as straight lines on a cartesian plane.
A standard linear equation is typically written as:

• \( Ax + By = C \)

This showcases relationships where the sum of the terms produces a constant output. Linear equations are particularly important because they help us understand relationships with a constant rate of change.
In slope-intercept form, an equation takes a different expression that's more intuitive for graphing, primarily focusing on the slope and y-intercept of the line. This is especially useful when we need to find quick information about how lines behave with different inputs.

Slope

The slope of a line in the context of linear equations denotes the steepness or incline. It is a critical component as it defines how the line moves on the graph relative to the x-axis. The slope is noted by the letter "\( m \)" in the slope-intercept form.
Mathematically, the slope is the "rise over run," meaning it is the ratio of the vertical change to the horizontal change between two points on the line:

• \( m = \frac{\text{change in } y}{\text{change in } x} \)

A positive slope means the line is moving upwards, while a negative slope indicates it's moving downwards. Zero slope results in a horizontal line, suggesting no vertical change regardless of the x-axis movement. With the provided example of \( m=-2 \), the line is purely downward sloping, signifying a decrease in output (y-value) for an increase in input (x-value).

Y-Intercept

The y-intercept of a line is a point at which the line crosses the y-axis. It provides invaluable information as it demonstrates the output value when the input (x-value) is zero. In the slope-intercept equation \( y = mx + b \), "\( b \)" depicts this crucial intercept.
Consider it the starting point of the line and is where all calculations begin when graphing a line on a cartesian plane. For instance, if the y-intercept is \( \frac{7}{3} \), as in our example, the line intersects the y-axis at this fractional point. This means that when \( x = 0 \), \( y \) will equal \( \frac{7}{3} \).
The y-intercept helps to easily graph the line by providing a fixed starting point, ensuring smoother calculations and visual representation when combined with the slope.