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Chapter 1: Problem 24

Finding Equations of Lines Find an equation of the line that satisfies the given conditions. Slope \(\frac{2}{5} ; \quad y\)-intercept 4

Short Answer

Expert verified
The line's equation is \( y = \frac{2}{5}x + 4 \).

Step by step solution

01

Identify the Formula for a Line Equation

To find the equation of a line, we use the slope-intercept form of the line equation: \( y = mx + b \). Here, \( m \) represents the slope of the line and \( b \) represents the y-intercept.
02

Substitute the Given Slope

We know from the problem that the slope \( m \) is \( \frac{2}{5} \). Substitute \( m = \frac{2}{5} \) into the formula. This gives us \( y = \frac{2}{5}x + b \).
03

Substitute the Given Y-Intercept

The problem states that the y-intercept \( b \) is 4. Substitute \( b = 4 \) into the equation. This modifies the equation to \( y = \frac{2}{5}x + 4 \).
04

Finalize the Line Equation

Now that we have substituted both the slope and y-intercept, the equation of the line is finalized as \( y = \frac{2}{5}x + 4 \).

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Key Concepts

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

Slope-Intercept Form
One of the simplest and most commonly used forms to express the equation of a line is the slope-intercept form. It is expressed as \( y = mx + b \).
Here, \( y \) represents the dependent variable or the output of the function for a given \( x \), which is the independent variable or the input. The form clearly shows the slope \( m \) and the y-intercept \( b \), making it easy to visualize and graph linear equations.

Using the slope-intercept form is advantageous because
  • It provides direct information about the steepness and direction of the line through the slope \( m \).
  • It clearly displays the point, \((0, b)\), where the line crosses the y-axis.
Understanding this form lays the groundwork for graphing lines and analyzing their behaviors.
Slope
Slope, denoted by \( m \) in the slope-intercept form, is a measure of the steepness and direction of a line. It represents the ratio of the vertical change (rise) to the horizontal change (run) between two points on the line.

Mathematically, this is given as:
\[m = \frac{\text{rise}}{\text{run}} = \frac{y_2 - y_1}{x_2 - x_1}\]
A positive slope means the line is rising from left to right, while a negative slope means the line is falling. A zero slope represents a horizontal line, and an undefined slope indicates a vertical line.
  • In our example, the slope is \( \frac{2}{5} \), indicating that for every five units we move horizontally, the line rises by two units.
  • Slope is crucial for determining how much \( y \) changes with a change in \( x \).
Y-Intercept
The y-intercept is the point where the line crosses the y-axis. In the slope-intercept form \( y = mx + b \), the y-intercept is represented by \( b \).
This tells us the value of \( y \) when \( x \) is zero, that is, the starting point of the line on the vertical axis. Thus, the y-intercept is instrumental in graphing the line, as it provides a fixed point through which the line passes.

For the exercise we examined:
  • The y-intercept is 4.
  • This means that when \( x = 0 \), \( y \) equals 4.
Graphically, this is where you would start plotting the line and then use the slope to find other points on the line.
Equation of a Line
An equation of a line is an algebraic expression that defines a straight line on a coordinate plane. It unifies the slope and y-intercept concepts we discussed.Lines communicate a constant rate of change, which is vital in a variety of fields, from physics to economics.The general procedure to find the equation of a line uses:
  • The slope \( m \), which we've set as \( \frac{2}{5} \) in our problem.
  • The y-intercept \( b \), here given as 4.
Using these in the formula \( y = mx + b \) gives us \( y = \frac{2}{5}x + 4 \), which explains the line's behavior entirely in a two-dimensional plane and allows you to graph it with ease. Each component, \( \frac{2}{5} \) and 4, plays a distinct role in forming the equation, offering insights into how the line behaves across different values of \( x \).

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Most popular questions from this chapter

Let \(a, b,\) and \(c\) be real numbers with \(a>0, b<0,\) and \(c<0 .\) Determine the sign of each expression. (a) \(b^{5}\) (b) \(b^{10}\) (c) \(a b^{2} c^{3}\) (d) \((b-a)^{3}\) (e) \((b-a)^{4}\) (f) \(\frac{a^{3} c^{3}}{b^{6} c^{6}}\)

Area of a Region Sketch the region in the coordinate plane that satisfies both the inequalities \(x^{2}+y^{2} \leq 9\) and \(y \geq|x| .\) What is the area of this region?

Manufacturer's Profit If a manufacturer sells \(x\) units of a certain product, revenue \(R\) and cost \(C\) (in dollars) are given by $$ \begin{array}{l} R=20 x \\ C=2000+8 x+0.0025 x^{2} \end{array} $$ Use the fact that profit \(=\) revenue \(-\) cost to determine how many units the manufacturer should sell to enjoy a profit of at least \(\$ 2400\).

Solve the equation for the variable \(x\). The constants \(a\) and \(b\) represent positive real numbers. $$\sqrt{x}-a \sqrt[3]{x}+b \sqrt[6]{x}-a b=0$$

Sketch the region given by the set. $$\left\\{(x, y) | x^{2}+y^{2}>4\right\\}$$
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