Problem 76 - Algebra 1: Concepts and Skills

Question

Write in slope-intercept form the equation of the line that passes through the given points. $$ (3,9) \text { and }(1,-3) $$

Step-by-Step Solution

Answer

The slope-intercept form of the equation of the line is \( y = 6x - 9 \).

Step 1: Find the Slope

Find the slope (m) of the line using the formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \). Substituting the given points into the formula, we have \( m = \frac{-3 - 9}{1 - 3} = 6. \)

Step 2: Write the Slope-Point Form

Using the point-slope form of the line, which is \( y - y_1 = m(x - x_1) \), with one of the given points and the slope, we get \( y - 9 = 6(x - 3) \).

Step 3: Convert to Slope-Intercept Form

Expanding and rearranging the equation from Step 2 to make y the subject, we get \( y = 6x - 9 \).

Key Concepts

Slope

Understanding the slope of a line is key to grasping how lines behave on a graph. The slope is a measure of how steep the line is. It tells us how much the line goes up or down as we move from left to right. In mathematical terms, the slope is represented by the letter \( m \) and is calculated using

• The change in y-values, \( y_2 - y_1 \), which is the vertical difference between two points.
• The change in x-values, \( x_2 - x_1 \), which is the horizontal difference.

The formula for the slope is \( m = \frac{y_2 - y_1}{x_2 - x_1} \). This formula helps find the rate at which y changes for every unit change in x.
In the example given, by substituting the points \((3, 9)\) and \((1, -3)\) into this formula, we compute the slope as \(6\). This tells us that for every 1 unit we move to the right, the line goes up by 6 units.

Point-Slope Form

The point-slope form of a line's equation comes into play once you know the slope of the line and at least one point through which it passes. It's a powerful tool for quickly building the equation of the line. The formula for point-slope form is:

• \( y - y_1 = m(x - x_1) \), where \((x_1, y_1)\) is a given point on the line.

This form emphasizes the relationship between changes in \( x \) and changes in \( y \), dictated by the slope \( m \).
In our exercise, using the point \((3, 9)\) and slope \(6\), the point-slope form becomes \( y - 9 = 6(x - 3) \). This setup directly links the coordinates of the point and the slope to form a preliminary equation of the line.

Equation of a Line

An equation of a line can be expressed in several forms, with the slope-intercept form being one of the most popular. This form directly shows both the slope of the line and where the line crosses the y-axis, known as the y-intercept.

• The general expression for this form is \( y = mx + b \), where \( b \) represents the y-intercept.

Converting from point-slope to slope-intercept form involves simple algebraic manipulation. Adjusting the equation from \( y - 9 = 6(x - 3) \), we expand and reorganize to isolate \( y \). This yields \( y = 6x - 9 \). Now, we're presented with an equation where \(6\) is the slope, and the line intersects the y-axis at \(-9\).
The process of converting between different line equation forms like these solidifies your understanding and helps in graphing the line accurately.