Problem 41 - Applied Calculus

Question

Write an equation of the line satisfying the following conditions. If possible, write your answer in the form \(y=m x+b\). Passing through the points (5,3) and (7,-1)

Step-by-Step Solution

Answer

The equation is \(y = -2x + 13\).

Step 1: Understand the Problem

We need to find the equation of the line that passes through the given points (5,3) and (7,-1). The equation should be in the slope-intercept form, which is \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.

Step 2: Calculate the Slope

First, we calculate the slope \(m\) of the line using the formula for slope \(m = \frac{y_2 - y_1}{x_2 - x_1}\). Substituting the points (5, 3) and (7, -1) gives:\[m = \frac{-1 - 3}{7 - 5} = \frac{-4}{2} = -2\\]So, the slope \(m\) is \(-2\).

Step 3: Find the Y-intercept

Now that we have the slope, we can use one of the points to find the y-intercept \(b\). Using the point (5,3) in the equation \(y = mx + b\):\[3 = -2(5) + b\]Simplifying, we get:\[3 = -10 + b \b = 13\]Thus, the y-intercept \(b\) is 13.

Step 4: Write the Equation

With the slope \(m = -2\) and the y-intercept \(b = 13\), the equation of the line is:\[y = -2x + 13\]This is the line that passes through the points (5,3) and (7,-1).

Key Concepts

Equation of a Line

Understanding how to write an equation of a line is an essential algebraic skill. The most commonly used form for a linear equation is the slope-intercept form, which is expressed as:\[ y = mx + b \]In this equation:

• \( y \) represents the dependent variable, or what you calculate based on \( x \)
• \( x \) is the independent variable, or the input
• \( m \) denotes the slope of the line, describing its steepness and direction
• \( b \) is the y-intercept, where the line crosses the y-axis

By knowing two points that the line passes through, you can derive both the slope \( m \) and the intercept \( b \), allowing you to write the complete line equation. This form is very handy because it gives you direct insight into two key properties of the linear relationship: slope and y-intercept. This understanding serves as a foundation for solving many real-world problems involving lines.

Slope Calculation

The slope of a line is a measure of its steepness and direction, and it is calculated by taking the vertical change (rise) and dividing it by the horizontal change (run) between two points on the line.The formula to calculate the slope \( m \) is:\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]Where:

• \( (x_1, y_1) \) and \( (x_2, y_2) \) are the coordinates of two different points on the line.

For example, consider points (5, 3) and (7, -1). The slope is calculated as:\[ m = \frac{-1 - 3}{7 - 5} = \frac{-4}{2} = -2 \]This means the line falls 2 units for every 1 unit it moves to the right (since the slope is negative). Grasping how to compute a line's slope helps you understand the relationship between the variables and predict how changes to one variable affect the other.

Y-intercept

The y-intercept is a critical component of the slope-intercept form \( y = mx + b \). It is the point where the line crosses the y-axis. To find the y-intercept \( b \), you can use the slope \( m \) along with one of the line’s points. Here's how you calculate it:Using the point-slope form, rearrange the slope-intercept formula:\[ y = mx + b \]Assuming you know a point \( (x_1, y_1) \) and the slope \( m \), solve for \( b \):\[ y_1 = mx_1 + b \]\[ b = y_1 - mx_1 \]For instance, using point (5, 3) and slope \( m = -2 \):\[ 3 = -2(5) + b \]\[ 3 = -10 + b \]\[ b = 13 \]So, the y-intercept \( b \) is 13. This indicates where the line meets the y-axis, providing a starting point for drawing the line when sketching graphs. Recognizing the importance of the y-intercept aids in interpreting real-world data points in linear models.