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Chapter 3: Problem 72

Use either the slope-intercept form (from Section 3.5) or the point-slope form (from Section 3.6) to find an equation of each line. Write each result in slope-intercept form, if possible. Slope \(\frac{9}{8},\) passes through the origin

Short Answer

Expert verified
The equation of the line is \( y = \frac{9}{8}x \). It is already in slope-intercept form.

Step by step solution

01

Identify Given Information

We are given the slope of the line as \( \frac{9}{8} \) and the line passes through the origin (0,0).
02

Use the Point-Slope Form

The point-slope form of a line is given by \( y - y_1 = m(x - x_1) \), where \( m \) is the slope and \((x_1, y_1)\) is the given point. Here, \( m = \frac{9}{8} \) and \((x_1, y_1) = (0, 0)\).
03

Substitute Values into Point-Slope Form

Substitute the given slope and point into the point-slope form: \( y - 0 = \frac{9}{8}(x - 0) \).
04

Simplify the Equation

Simplify the equation to get \( y - 0 = \frac{9}{8}x \) or \( y = \frac{9}{8}x \).
05

Convert to Slope-Intercept Form

The equation \( y = \frac{9}{8}x \) is already in the slope-intercept form \( y = mx + b \) where \( m \) is the slope and \( b \) is the y-intercept (\( b = 0 \) since it passes through the origin).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Slope-Intercept Form
The slope-intercept form of a linear equation is one of the most useful and popular ways to express lines on a graph. The standard form of this equation is written as:
  • \[ y = mx + b \]
Here, \( m \) represents the slope of the line, and \( b \) denotes the y-intercept, which is where the line crosses the y-axis. This form makes it easy to identify both the slope and the y-intercept of any line, making graphing quick and straightforward.

For example, consider the line equation \( y = \frac{9}{8}x \). This may seem simple, but it tells us a lot:
  • The slope \( m \) is \( \frac{9}{8} \), meaning for every 8 units you move horizontally to the right, you move 9 units up.
  • The y-intercept \( b \) is 0, which indicates the line passes through the origin.
Point-Slope Form
The point-slope form is a great alternative when you know a point on the line and the slope. It is written as:
  • \[ y - y_1 = m(x - x_1) \]
In this formula, \((x_1, y_1)\) is a specific point on the line, and \( m \) is the slope. This kind of equation is helpful when constructing the equation of a line when you're provided with specific points.

In the given exercise, the line passes through the origin, so our point is \((0, 0)\). This turns our equation into a really simple form:
  • Set \( x_1 = 0 \), \( y_1 = 0 \), and \( m = \frac{9}{8} \), giving us \[ y - 0 = \frac{9}{8}(x - 0) \].
Upon simplifying, this can then be directly converted to the slope-intercept form, revealing the direct relationship between the two types.
Slope Calculation
Calculating the slope is a foundational skill in algebra, crucial for understanding how lines and linear equations work. Slope is the measure of steepness or the incline of a line.
To find a slope using two points \((x_1, y_1)\) and \((x_2, y_2)\), you use the formula:
  • \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
The formula calculates the change in y-coordinates (vertical change) over the change in x-coordinates (horizontal change). A positive slope means the line rises as you move to the right, while a negative slope indicates it falls.

In our exercise, the slope is given as \( \frac{9}{8} \), which tells us:
  • For every 8 units you move to the right, you rise 9 units vertically.
This straightforward method makes relationships between points and lines easy to comprehend.
Origin in Algebra
In algebra, the term "origin" refers to the point on a graph where both the x-coordinate and the y-coordinate are zero. It is represented by the coordinates \((0, 0)\). The origin is a special point because it serves as the intersection of the x-axis and y-axis, essentially dividing the entire graph area into four equal quadrants.
Why is the origin significant?
  • It provides a clear starting point for plotting graphs, offering a consistent reference across all Cartesian planes.
  • For any line passing through the origin, the y-intercept \(b\) is zero, simplifying the line's equation to \(y = mx\).
In our case, because the line passes through \((0, 0)\), it simplifies both the point-slope and slope-intercept forms of the equation, making it easy and quick to solve.

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Most popular questions from this chapter

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