Problem 27 - College Algebra with Modeling and Visualization

Question

Find the slope-intercept form for the line satisfying the conditions. y-intercept \(45,\) x-intercept 90

Step-by-Step Solution

Answer

The slope-intercept form of the line is \( y = -\frac{1}{2}x + 45 \).

Step 1: Understand Slope-Intercept Form

The slope-intercept form of a line is given by \( y = mx + c \), where \( m \) is the slope and \( c \) is the y-intercept. We are given the y-intercept as 45.

Step 2: Identify the Given Points

The y-intercept is given as \( (0, 45) \) and the x-intercept is given as \( (90, 0) \). These are the points where the line crosses the y-axis and x-axis, respectively.

Step 3: Calculate the Slope

The slope \( m \) of the line can be found using the formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \) where \((x_1, y_1)\) and \((x_2, y_2)\) are two points on the line. Substitute the intercepts: \((0, 45)\) and \((90, 0)\). Hence, \( m = \frac{0 - 45}{90 - 0} = \frac{-45}{90} = -\frac{1}{2}\).

Step 4: Write the Equation

Now that we have the slope \( m = -\frac{1}{2} \) and the y-intercept \( c = 45 \), substitute these values into the slope-intercept form. Thus, the equation is \( y = -\frac{1}{2}x + 45 \).

Key Concepts

Linear Equations

Linear equations are fundamental in mathematics, representing a straight line on a graph. These equations take the form \( y = mx + c \), which is commonly known as the slope-intercept form. Here, \( y \) and \( x \) are variables, while \( m \) and \( c \) are constants. The main features of a linear equation include the slope and the y-intercept, which uniquely determine the line's position and direction in the coordinate plane.
Linear equations can model various real-world situations, such as predicting expenses, analyzing trends, or finding constant rates of change. Because of their utility, it's crucial to understand how to work with these types of equations. They involve constant changes between \( x \) and \( y \), making them excellent predictors. Recognizing and constructing linear equations helps solve practical problems efficiently.

y-intercept

The y-intercept is a key component in the equation of a line. It refers to the point where the line crosses the y-axis. In the slope-intercept form \( y = mx + c \), \( c \) represents the y-intercept. This means that when \( x = 0 \), \( y = c \).
For example, if the y-intercept is 45, the line crosses the y-axis at the point \( (0, 45) \). This point is crucial because it provides a starting point to draw the line even if the whole equation isn't known yet.
Understanding the y-intercept helps in sketching the graph of a line and serves as a constant in the equation regardless of the slope. Whether you're analyzing graphs or solving problems, knowing the y-intercept simplifies the process.

x-intercept

The x-intercept is another important feature in the context of linear equations. It is the point where the line crosses the x-axis. To find it, one can set \( y = 0 \) in the equation and solve for \( x \).
In our example, the x-intercept is given as 90, which means the line crosses the x-axis at the point \( (90, 0) \). This provides critical information about the line's path through the coordinate plane.
Knowing both intercepts (x and y) is particularly useful in visualizing the line and understanding its behavior. The intercepts help define the boundaries of the graph, making it easier to interpret the line's movement and connections within the graph.

Slope Calculation

Calculating the slope of a line accurately is vital in understanding its steepness and direction. The slope, represented by \( m \) in the equation \( y = mx + c \), measures the change in \( y \) for every unit change in \( x \).
The formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \) helps calculate the slope given two points on the line. For our problem, we use the points \((0, 45)\) and \((90, 0)\). Substitute these into the formula to get \( m = \frac{0 - 45}{90 - 0} = -\frac{1}{2} \). This result means that for every 2 units increase in \( x \), \( y \) decreases by 1 unit.
Understanding how to calculate the slope provides insights into the line's angle and growth rate. A positive slope means rising, while a negative one means falling, and zero indicates a flat horizontal line. The knowledge of slope is essential for analyzing trends and motions in mathematics and real-world applications.