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Chapter 1: Problem 27

Find the slope and y-intercept (if possible) of the equation of the line. $$ x+5 y=20 $$

Short Answer

Expert verified
The slope of the line is -1/5 and the y-intercept of the line is 4.

Step by step solution

01

Rewrite the Equation in Slope-Intercept Form

To convert the given equation into slope-intercept form, it needs to be rewritten as \(y = mx + b\). Currently x and y are on the same side, so they need to be separated. This means subtracting \(x\) from both sides of the equation. So, the equation becomes: \(5y = -x + 20\). Then, to completely isolate \(y\), both sides will be divided by 5. This results in: \(y = -\frac{1}{5}x + 4\).
02

Identify the Slope

The slope (m) of the equation can be identified by looking at the coefficient of \(x\) when the equation is written in the form \(y = mx + b\). In this case, the slope \(m\) is -1/5.
03

Identify the y-intercept

The y-intercept (b) of the equation is the constant term when the equation is written in slope-intercept form. It's the point where the line crosses the y-axis. In this case, the y-intercept \(b\) is 4.

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Key Concepts

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

Slope-Intercept Form
In algebra, the slope-intercept form of a linear equation is super handy for describing straight lines. It's written as \(y = mx + b\). Here, "\(m\)" represents the slope of the line, and "\(b\)" is the y-intercept.

To begin, let's talk about why this form is useful. The slope-intercept form makes it easy to quickly understand a line's most important features: how steep the line is and where it cuts through the vertical axis (the y-axis). This quick readability is a big plus when graphing lines or solving problems related to them.
  • "\(y\)" is the dependent variable and represents the line's value at any point along the x-axis.
  • "\(x\)" is the independent variable, showcasing the input or the domain of the function.
  • The equation is crucial in providing a straightforward method to sketch or interpret a line on a graph.

This form is particularly useful in identifying the slope and y-intercept, which we will dive deeper into below.
Slope of a Line
Now, let's understand the slope of a line, which tells us how steep the line is. Think of the slope as the 'tilt' of the line, indicating whether it moves upwards or downwards as it progresses.

In the equation \(y = -\frac{1}{5}x + 4\), the slope \(m\) is \(-\frac{1}{5}\). Here's what that means:
  • A negative slope, like \(-\frac{1}{5}\), means the line slants downwards as it moves from left to right.
  • A slope value of \(-\frac{1}{5}\) specifically indicates that for every unit increase in "\(x\)", "\(y\)" decreases by \(\frac{1}{5}\) of a unit.
  • The steeper the slope, the faster it rises or falls.

Understanding the slope is crucial for graphing, predicting changes, and interpreting relationships between variables in linear equations.
Y-Intercept
The y-intercept is where the line crosses the y-axis. It's like the starting point of the line on a graph, assuming "\(x = 0\)". For our equation \(y = -\frac{1}{5}x + 4\), the y-intercept \(b\) is 4.
  • The y-intercept is expressed as a point on the coordinate plane, so in this case, it's the point \((0, 4)\).
  • It tells us that when "\(x = 0\)", "\(y\)" equals 4. This is a key reference point for graphing.
  • Knowledge of the y-intercept allows us to immediately plot where the line begins on the y-axis, aiding in drawing the rest of the line based on its slope.

The y-intercept gives an intuitive start to understanding the behavior of the entire function on a graph.

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Most popular questions from this chapter

Describe the interval(s) on which the function is continuous. $$ f(x)=x \sqrt{x+3} $$

use a graphing utility to graph the function. Then use the Horizontal Line Test to determine whether the function is one-to-one. If it is, find its inverse function. $$ g(x)=x^{2} \sqrt{x^{2}-4} $$

Market Equilibrium The supply function for a product relates the number of units \(x\) that producers are willing to supply for a given price per unit \(p .\) The supply and demand functions for a market are $$ \begin{array}{l}{p=\frac{2}{5} x+4} \\ {p=-\frac{16}{25} x+30}\end{array} $$ (a) Use a graphing utility to graph the supply and demand functions in the same viewing window. (b) Use the trace feature of the graphing utility to find the equilibrium point for the market. (c) For what values of does the demand exceed the supply? (d) For what values of does the supply exceed the demand?

use a graphing utility to graph the function. Then use the zoom and trace features to find the zeros of the function. Is the function one-to-one? $$ f(x)=9 x-4 x^{2} $$

Find the limit. \(\lim _{x \rightarrow 2}\left(-x^{2}+x-2\right)\)
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