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Chapter 0: Problem 34

Find an equation of the line: with \(y\) -intercept (0,7) and slope \(\frac{4}{3}\).

Short Answer

Expert verified
The equation of the line is \( y = \frac{4}{3}x + 7 \).

Step by step solution

01

Identify Line Equation Format

The general format for a linear equation is the slope-intercept form: \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.
02

Insert the Given Slope

Replace \( m \) in the equation with the given slope \( \frac{4}{3} \). The equation becomes \( y = \frac{4}{3}x + b \).
03

Use the Y-Intercept

The y-intercept is given as \( (0,7) \), meaning when \( x = 0 \), \( y = 7 \). Therefore, \( b = 7 \).
04

Write the Complete Equation

Substitute \( b = 7 \) into the equation from Step 2: \( y = \frac{4}{3}x + 7 \). This is the equation of the line you are finding.

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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 is a fundamental concept in algebra that describes the equation of a straight line. Represented as \( y = mx + b \), this form makes it easy to understand how different variables are related in a linear equation. Here:
  • \( y \) is the dependent variable, reliant on the values of \( x \).
  • \( m \) stands for the slope of the line.
  • \( b \) is the y-intercept, where the line crosses the y-axis.
To graph a line using this form, start by plotting the y-intercept \( (0, b) \) on the Cartesian plane. From there, use the slope \( m \), expressed as a fraction \( \frac{rise}{run} \), to determine the next points on the line. This method is convenient because you can quickly visualize how changes in the slope or y-intercept affect the position and angle of a line.
Slope
The slope of a line is a measure of its steepness. In the slope-intercept form, the slope is denoted by \( m \). A positive slope means the line ascends from left to right, while a negative slope means it descends. Calculating the slope involves the change in y divided by the change in x, often referred to as "rise over run."
  • For instance, a slope of \( \frac{4}{3} \) means that鈥攆or every 3 units moved to the right (run), the line rises by 4 units (rise).
  • The units of the slope intention change the nature of the relationship between variables. The higher the absolute value of \( m \), the steeper the line.
Understanding slope is crucial for analyzing how much one variable changes in response to another, especially in contexts like speed, velocity, or rates of change.
Y-Intercept
The y-intercept is a key feature of a linear equation in the slope-intercept form and is symbolized as \( b \) in \( y = mx + b \). It represents the point where the line crosses the y-axis, indicating the value of \( y \) when \( x = 0 \). For example:
  • A y-intercept of 7 suggests that the line crosses the y-axis at the point (0,7).
  • The y-intercept affects the vertical positioning of the line on the graph.
This feature is particularly helpful because it gives a starting point for drawing the line on a graph without needing to calculate other values first. Changes in the y-intercept will move the line up or down along the y-axis, which can be used to model real-world situations where initial values matter, such as starting conditions in financial or scientific models.

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