Problem 23 - Calculus: One and Several Variables

Question

Write an equation for the line with slope -5 and \(y\) -intercept 2

Step-by-Step Solution

Answer

The equation of the line is \(y = -5x + 2\)

Step 1: Understanding the question

The question is asking to write down the equation of the line in the form of \(y = mx + b\) where \(m\) (slope of the line) is -5 and \(b\) (y-intercept of the line) is 2.

Step 2: Substituting the values into the equation

Now we substitute \(m = -5\) and \(b = 2\) into the formula. Therefore, the equation of the line is \(y = -5x + 2\).

Key Concepts

Slope

The concept of slope is essential when discussing linear equations and lines in general. Slope refers to how steep a line is and can be thought of as the "rise over run." It tells us how much the line rises (or falls) for a given horizontal move to the right. Mathematically, slope is represented by the letter \( m \). To find the slope between two points \((x_1, y_1)\) and \((x_2, y_2)\), you use the formula

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

A positive slope means the line ascends as you move from left to right, while a negative slope means it descends. If the slope is zero, the line is perfectly horizontal. In the exercise, the given slope is -5, indicating a descending line as you move left to right. This negative value means for every unit you move to the right, the line moves 5 units down.
Understanding slope helps in determining how two different lines may interact, such as being parallel (same slope) or intersecting (different slopes).

Y-Intercept

The y-intercept is where a line crosses the y-axis on a graph. This point is crucial because it gives a starting point for the line. In the standard form of a line equation \( y = mx + b \), the y-intercept is represented by \( b \). When \( x = 0 \), the value of \( y \) is equal to \( b \), indicating the point at which the line meets the y-axis.
For example, if a line has a y-intercept of 2, this means that the line crosses the y-axis at \( y = 2 \). Graphically, if you were to draw the line, you'd start at this point.
The y-intercept can dramatically affect the positioning of a line on a graph. Two lines with the same slope but different y-intercepts will be parallel to one another.
Comprehending the y-intercept allows us to more predictably graph lines and understand the initial point of contact with the y-axis.

Linear Equations

Linear equations are equations that make a straight line when graphed. They are one of the simplest and most important forms of equations in algebra. The general form of a linear equation is \( y = mx + b \), where \( m \) represents the slope and \( b \) represents the y-intercept.
Linear equations are characterized by constant rates of change. This means that as you move along the line, the relationships between the variables do not change, producing proportional outputs for given inputs. This is why lines have uniform directions without curves.
Linear equations are used to model real-world situations where the rate of change is constant, such as calculating interest or predicting costs.
In our exercise, the linear equation resulting from given values is \( y = -5x + 2 \). This equation tells us precisely how to draw the line on a graph by starting from the y-intercept (2) and using the slope (-5) to determine its angle and direction. Understanding linear equations simplifies many mathematical and real-world problems, as knowing just the slope and y-intercept, as seen in the exercise, is enough to determine the entire function.