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Chapter 1: Problem 2

Find the slopes and \(y\)-intercepts of the following lines. $$y=\frac{3 x+1}{5}$$

Short Answer

Expert verified
The slope is $$\frac{3}{5}$$ and the $$y$$-intercept is $$\frac{1}{5}.$$

Step by step solution

01

Identify the given equation

The equation provided is given in the form $$y=\frac{3 x+1}{5}$$
02

Rewrite the equation in slope-intercept form

The slope-intercept form of a linear equation is given by $$y = mx + b$$ where $$m$$ is the slope and $$b$$ is the $$y$$-intercept. The given equation can be rewritten by distributing the fraction: $$y = \frac{3}{5}x + \frac{1}{5}$$
03

Determine the slope

From the rewritten equation, $$y = \frac{3}{5}x + \frac{1}{5},$$ the coefficient of $$x$$ is the slope. Therefore, the slope ($$m$$) is $$\frac{3}{5}.$$
04

Determine the $$y$$-intercept

In the equation $$y = \frac{3}{5}x + \frac{1}{5},$$ the constant term is the $$y$$-intercept. Hence, the $$y$$-intercept ($$b$$) is $$\frac{1}{5}.$$

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Key Concepts

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

Linear Equations
Linear equations are equations that make a straight line when graphed on a coordinate plane. They have the general form:
\[y = mx + b\] where:
  • \(y\) is the dependent variable,
  • \(x\) is the independent variable,
  • \(m\) is the slope,
  • \(b\) is the y-intercept.
Linear equations are very useful in various fields such as science, engineering, and finance because they help describe relationships between two quantities. For example, if you know how much you earn per hour (\(m\)) and your base pay (\(b\)), you can write a linear equation to find out your total salary (\(y\)) for any amount of hours worked (\(x\)).
Slope
The slope is a fundamental concept in linear equations. It represents the rate of change between the dependent and independent variables. In the equation \(y = mx + b\), the slope (\(m\)) determines how steep the line is.
  • A positive slope means the line rises as you move from left to right.A negative slope means the line falls as you move from left to right.A slope of zero means the line is horizontal, showing no change in y as x changes.
Calculating the slope is fairly simple. For instance, in the equation \(y = \frac{3}{5}x + \frac{1}{5}\), the slope is \(\frac{3}{5}\). This tells us for every one unit increase in \(x\), \(y\) increases by \(\frac{3}{5}\) units. Understanding the slope helps interpret how changes in one variable affect another.
Y-Intercept
The y-intercept of a linear equation is the point where the line crosses the y-axis. In the equation \(y = mx + b\), the y-intercept is represented by \(b\). It tells us the value of \(y\) when \(x = 0\). For example, in the rewritten equation \(y = \frac{3}{5}x + \frac{1}{5}\), the y-intercept \(\frac{1}{5}\) signifies that when there is no input (\(x = 0\)), the output (\(y\)) is \(\frac{1}{5}\).
  • Knowing the y-intercept helps us quickly plot the starting point of the line on the graph.
  • It also provides a baseline value for the dependent variable when predicting outcomes.
The y-intercept is essential for understanding the initial condition of the relationship described by the linear equation.

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Most popular questions from this chapter

Use limits to compute \(f^{\prime}(x) .\) $$f(x)=3 x+1$$

After inspecting a sunken ship at a depth of 212 feet, a diver starts her slow ascent to the surface of the ocean, rising at the rate of 2 feet per second. Find \(y(t),\) the depth of the diver, measured in feet from the ocean's surface, as a function of time \(t\) (in seconds).

Use limits to compute the following derivatives. $$f^{\prime}(3), \text { where } f(x)=x^{2}+1$$

Write the equation of the tangent line to the curve \(y=x^{3}+3 x-8\) at \((2,6).\)

Let \(f(x)\) be the number of toys sold when \(x\) dollars are spent on advertising. Interpret the statements \(f(100,000)=3,000,000\) and \(f^{\prime}(100,000)=30\).
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