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Chapter 3: Problem 52

Find an equation of each line described. Write each equation in slope- intercept form when possible. Through (4,7) and (0,0)

Short Answer

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

Step by step solution

01

Understanding the Problem

We need to find the equation of a line that passes through the points (4,7) and (0,0), and express the equation in slope-intercept form, if possible.
02

Finding the Slope

Calculate the slope of the line using the formula for the slope \( m = \frac{y_2 - y_1}{x_2 - x_1} \). Here, \( (x_1, y_1) = (4, 7) \) and \( (x_2, y_2) = (0, 0) \). Substitute these into the formula: \( m = \frac{0 - 7}{0 - 4} = \frac{-7}{-4} = \frac{7}{4} \).
03

Using Point-Slope Form

Once we have the slope \( m = \frac{7}{4} \), use the point-slope form of the equation: \( y - y_1 = m(x - x_1) \). Substitute \( m \), \( x_1 = 4 \), and \( y_1 = 7 \) into the equation: \( y - 7 = \frac{7}{4}(x - 4) \).
04

Converting to Slope-Intercept Form

The equation from the previous step is \( y - 7 = \frac{7}{4}(x - 4) \). Distribute \( \frac{7}{4} \): \( y - 7 = \frac{7}{4}x - 7 \). Add 7 to both sides to solve for \( y \): \( y = \frac{7}{4}x \). This is the slope-intercept form \( y = mx + b \) with \( m = \frac{7}{4} \) and \( b = 0 \).

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Key Concepts

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

Equation of a Line
An equation of a line provides a mathematical representation of a straight line on a coordinate plane. It is essential in algebra and geometry for describing the relationship between any two points along that line.

Generally, the most common form representing the equation of a line is the slope-intercept form, expressed as:
  • \( y = mx + b \)
Here:
  • \( y \) is the dependent variable (typically the output).
  • \( x \) is the independent variable (typically the input).
  • \( m \) represents the slope of the line, which indicates the line's steepness.
  • \( b \) is the y-intercept, the point where the line crosses the y-axis.
Understanding and using line equations enables us to predict and comprehend the behavior of linear relationships. This can be useful in fields such as economics, physics, and biology. Knowing how to convert line equations between different forms, like from point-slope form to slope-intercept form, is a valuable skill.
Slope Calculation
The slope of a line is a measure of its steepness and is represented by the letter \( m \). It tells us how much \( y \) changes for a given change in \( x \).

The formula for calculating the slope between two points \((x_1, y_1)\) and \((x_2, y_2)\) is:
  • \( m = \frac{y_2 - y_1}{x_2 - x_1} \)
This difference quotient gives us the ratio of vertical change (rise) to horizontal change (run) between two points.

Let's consider an example: to calculate the slope between the points (4, 7) and (0, 0), apply the formula:
  • \( m = \frac{0 - 7}{0 - 4} = \frac{-7}{-4} = \frac{7}{4} \)
In this example, the slope \( \frac{7}{4} \) indicates that for every 4 units you move horizontally to the right, the line rises 7 units. This positive slope indicates an upward incline from left to right.
Point-Slope Form
The point-slope form is another way of expressing the equation of a line. It is particularly useful when you know a point on the line and the slope.

The formula for the point-slope form is:
  • \( y - y_1 = m(x - x_1) \)
Here:
  • \((x_1, y_1)\) is a known point on the line.
  • \( m \) is the slope of the line.
For example, if a line passes through the point (4, 7) with a slope of \( \frac{7}{4} \), the point-slope form would be:
  • \( y - 7 = \frac{7}{4}(x - 4) \)
Using this form helps you directly involve known points and slopes, making it easy to derive the equation in other forms.

By rearranging the equation, you can transform the line from the point-slope form to the slope-intercept form of \( y = mx + b \). This transformation is crucial when you need to easily identify slope and y-intercept for graphing or further applications.

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Most popular questions from this chapter

Complete each ordered pair so that it is a solution of the given linear equation. See Example 5. $$ y=\frac{1}{5} x-2 ;(-10, \quad),(\quad, 1) $$

Complete each ordered pair so that it is a solution of the given linear equation. See Example 5. $$ y=\frac{1}{4} x-3 ;(-8, \quad),(\quad, 1) $$

The amount \(y\) of land occupied by farms in the United States (in millions of acres) from 1997 through 2007 is given by \(y=-4 x+967\). In the equation, \(x\) represents the number of years after 1997 . (Source: National Agricultural Statistics Service) a. Complete the table. $$ \begin{array}{|c|c|c|c|} \hline \boldsymbol{x} & 4 & 7 & 10 \\ \hline \boldsymbol{y} & & & \\ \hline \end{array} $$ b. Find the year in which there were approximately 930 million acres of land occupied by farms. (Hint: Find \(x\) when \(y=930\) and round to the nearest whole number.) c. Use the given equation to predict when the land occupied by farms might be 900 million acres. (Use the hint for part b.)

Solve each equation for y. See Section 2.5. $$ x+y=5 $$

Determine whether (1,1) is included in each graph. $$ y>5 x $$
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