Problem 23 - Calculus Single Variable

Question

Write the slope-intercept equation of the line that passes through the two given points. $$ (-1,3),(2,-9) $$

Step-by-Step Solution

Answer

The equation is \( y = -4x - 1 \).

Step 1: Understand the Slope-Intercept Form

The slope-intercept form of a line is given by the equation \( y = mx + b \), where \( m \) is the slope of the line and \( b \) is the y-intercept. Our goal is to find these values.

Step 2: Find the Slope (m)

The formula to find the slope \( m \) of a line passing through two points \((x_1, y_1)\) and \((x_2, y_2)\) is \( m = \frac{y_2 - y_1}{x_2 - x_1} \). Given points \((-1,3)\) and \((2,-9)\), calculate \( m \) as follows: \( m = \frac{-9 - 3}{2 - (-1)} = \frac{-12}{3} = -4 \).

Step 3: Use One Point to Find the Y-Intercept (b)

Select one of the points, e.g., \((-1, 3)\), and use it to find \( b \). Substitute \( m = -4 \), \( x = -1 \), and \( y = 3 \) into the equation \( y = mx + b \) to find \( b \): \( 3 = -4(-1) + b \). Solve for \( b \): \( 3 = 4 + b \) which gives \( b = -1 \).

Step 4: Write the Final Equation

Now that we know the slope \( m = -4 \) and the y-intercept \( b = -1 \), substitute these values into the slope-intercept form equation: \( y = -4x - 1 \). Therefore, the equation of the line is \( y = -4x - 1 \).

Key Concepts

Linear Equations

Linear equations are fundamental tools in mathematics, representing lines through algebraic expressions. They can be expressed in various forms, but the slope-intercept form is particularly handy for immediately identifying two crucial features of a line – the slope and the y-intercept.

In the slope-intercept form, the equation of a line is written as \( y = mx + b \). Here:

• \( y \) represents the dependent variable, typically plotted on the vertical axis.
• \( x \) stands for the independent variable on the horizontal axis.
• \( m \) is the slope of the line, showing its steepness.
• \( b \) denotes the y-intercept, indicating where the line crosses the y-axis.

Understanding linear equations and how to express them using the slope-intercept form is essential for graphing lines and analyzing their characteristics in both mathematical and real-world contexts.

Finding the Slope

Finding the slope of a line involves determining its rate of change in the vertical direction relative to a change in the horizontal direction.

The slope, symbolized as \( m \), is calculated from any two points on the line using the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] This formula works by measuring how much \( y \) changes for a one-unit change in \( x \). Each pair of points \((x_1, y_1)\) and \((x_2, y_2)\) uniquely determines the slope of a straight line.

For example, let's find the slope of a line going through two points: \((-1, 3)\) and \((2, -9)\). Using our formula:

• Subtract the y-values: \(-9 - 3 = -12\).
• Subtract the x-values: \(2 - (-1) = 3\).
• Calculate the slope: \(m = \frac{-12}{3} = -4\).

This slope of \(-4\) means that for every unit increase along the x-axis, y decreases by 4 units. This negative slope indicates a line falling from left to right when graphed.

Y-Intercept Calculation

The y-intercept is a critical aspect of understanding a line's equation, as it tells you where the line crosses the y-axis. To find it, we use one point on the line along with the slope.

Once you have the slope, substitute it, along with the coordinates of a known point, into the slope-intercept form equation to solve for the y-intercept \( b \). Using one of our points, say \((-1, 3)\), where \( m = -4 \):

• Start with the equation: \( y = mx + b \).
• Replace \( y \) with 3, \( x \) with -1, and \( m \) with -4.
• This gives us: \( 3 = -4(-1) + b \).
• Solve for \( b \): \( 3 = 4 + b \), so \( b = -1 \).

Thus, we determine that the y-intercept \( b \) is \(-1\). The full equation of the line becomes \( y = -4x - 1 \). This information is invaluable, as the y-intercept provides a starting point for graphing the line.