Problem 10 - Algebra 1

Question

Write an equation in slope-intercept form of the line that passes through the points. $$ (3,-2),(-6,4) $$

Step-by-Step Solution

Answer

The equation of the line in slope-intercept form that passes through the points (3,-2) and (-6,4) is \(y = \frac{2}{3}x - 4\).

Step 1: Calculate the Slope

The slope (m) between any two points \((x_1, y_1)\) and \((x_2, y_2)\) is calculated using the formula: \(m = \frac{y_2 - y_1}{x_2 - x_1}\). Substituting \((3, -2)\) into \((x_1, y_1)\) and \((-6, 4)\) into \((x_2, y_2)\), we get: \(m = \frac{4 - (-2)}{-6 - 3} = -\frac{6}{-9} = \frac{2}{3}\).

Step 2: Calculate the Y-intercept

After finding the slope, now substitute the slope \(\frac{2}{3}\) and one of the points \((3, -2)\) into our slope-intercept form \(y = mx + b\). From doing so, we can solve for \(b\). Thus, \(-2 = \frac{2}{3}(3) + b\). Solving for \(b\) we get \(b = -2 - 2 = -4\).

Step 3: Form the Equation of the Line

The slope-intercept form of the line is then \(y = mx + b\). Substitute \(m = \frac{2}{3}\) and \(b = -4\) into the equation. The final equation is \(y = \frac{2}{3}x - 4\)

Key Concepts

Equation of a Line

The equation of a line is a mathematical way to describe all the points that lie on a straight line. One of the most common forms for expressing a line's equation is the slope-intercept form. This is written as \( y = mx + b \), where \( m \) represents the slope of the line, and \( b \) denotes the y-intercept.

This form of an equation is quite handy because it directly tells us two critical characteristics of the line: the slope and where the line crosses the y-axis. Understanding these components helps in graphing the line and analyzing its behavior quickly.

The slope \( m \) indicates the steepness or inclination of the line, showing how much \( y \) increases or decreases as \( x \) increases by 1 unit. Meanwhile, the y-intercept \( b \) is the point where the line crosses the y-axis, giving us a starting point to sketch our line.

Slope Calculation

To calculate the slope, we use the formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \). This formula finds the difference in the y-values divided by the difference in the x-values between two points on the line. In easy terms, it's how much up or down the line goes for every step it takes to the right.

Consider two points on our line: (3, -2) and (-6, 4). Plugging into our formula:

• \( y_2 - y_1 = 4 - (-2) = 6 \)
• \( x_2 - x_1 = -6 - 3 = -9 \)

So, the slope \( m \) becomes \( \frac{6}{-9} = -\frac{2}{3} \). A negative slope indicates that as \( x \) increases, \( y \) decreases.

Understanding how to calculate the slope is vital because it not only helps in writing the equation of the line but also provides insights on the direction and style of the line's graph.

Y-Intercept

The y-intercept \( b \) is the point on a graph where the line crosses the y-axis. It's a pivotal part of the slope-intercept format \( y = mx + b \).

To find \( b \), substitute the known slope and a point from the line into your equation and solve for \( b \). In our example, using the slope \( \frac{2}{3} \) and the point (3, -2):

• Insert these into the equation \( y = mx + b \):
• \( -2 = \frac{2}{3}(3) + b \)
• \( -2 = 2 + b \)
• Solving gives us \( b = -4 \)

So, the line crosses the y-axis at the point (0, -4). The y-intercept provides a starting point for graphing and aids in quickly identifying one of the line's key attributes without performing complex calculations.