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Chapter 8: Problem 39

Write an equation of each line. Write the equation in standard form unless indicated otherwise. See Examples 1 through \(6 .\) Through (1,6) and (5,2)\(;\) use slope-intercept form.

Short Answer

Expert verified
The equation in slope-intercept form is \( y = -x + 7 \).

Step by step solution

01

Determine the slope

To find the slope of a line passing through two points, use the formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \). For the points (1,6) and (5,2), plug in these values to get \( m = \frac{2 - 6}{5 - 1} = \frac{-4}{4} = -1 \).
02

Use the slope-intercept form

The slope-intercept form of a line is \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. We have found \( m = -1 \). Substitute this value into the equation to get \( y = -x + b \).
03

Find the y-intercept

Use one of the given points to find \( b \). Using the point (1,6), substitute into \( y = mx + b \): \( 6 = -1(1) + b \). Solve for \( b \): \( 6 = -1 + b \), so \( b = 7 \).
04

Write the final equation

Substitute \( b = 7 \) back into \( y = -x + b \) to get the equation of the line. Thus, the equation is \( y = -x + 7 \).

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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 of a linear equation is a common way to express the equation of a line. It is written as \( y = mx + b \), where:
  • \( m \) represents the slope of the line
  • \( b \) represents the y-intercept, which is the point where the line crosses the y-axis
This form is particularly useful because it directly shows the y-intercept and the slope, allowing you to easily graph the line. To convert from slope-intercept form to another form, such as the standard form, simply rearrange the terms accordingly.
Standard Form of a Line
The standard form of a line is typically expressed as \( Ax + By = C \), where \( A \), \( B \), and \( C \) are integers, and \( A \) should be a non-negative integer. An essential characteristic of standard form is that it allows us to analyze both x- and y-intercepts more straightforwardly.
  • It emphasizes the coefficients of both variables in an equation.
  • Useful for solving systems of linear equations, where you need the equations to have the variables aligned vertically.
To convert from slope-intercept form (\( y = mx + b \)) to standard form, you will rearrange the equation so that both \( x \) and \( y \) terms are on one side and the constant is on the other side. This involves moving terms using addition or subtraction and eliminating any fractional coefficients by multiplying through by a common factor.
Calculating Slope
The slope is a measure of the steepness and direction of a line, and it's a core concept when working with linear equations. To calculate the slope \( m \) between two points \((x_1, y_1)\) and \((x_2, y_2)\), use the formula:\[m = \frac{y_2 - y_1}{x_2 - x_1}\]
  • If the slope is positive, the line rises as it moves from left to right.
  • If the slope is negative, like in our problem (\( m = -1 \)), the line falls as it moves from left to right.
  • A slope of zero indicates a horizontal line, while an undefined slope indicates a vertical line.
Calculating the slope accurately is key to forming the correct equation for a line, especially when you need to use the slope-intercept form or standard form. This process is also crucial for understanding how different lines interact and change in real-world applications.

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Most popular questions from this chapter

Suppose that a movie is being filmed in New York City. An action shot requires an object to be thrown upward with an initial velocity of 80 feet per second off the top of 1 Madison Square Plaza, a height of 576 feet. Neglecting air resistance, the height \(h(t)\) in feet of the object after \(t\) seconds is given by the function \(h(t)=-16 t^{2}+80 t+576 .\) (Source: The World Almanac, 2001 ) a. Find the height of the object at \(t=0\) seconds, \(t=2\) seconds, \(t=4\) seconds, and \(t=6\) seconds. b. Explain why the height of the object increases and then decreases as time passes. c. Factor the polynomial \(-16 t^{2}+80 t+576\).

If \(P(x)=3 x+3, Q(x)=4 x^{2}-6 x+3,\) and \(R(x)=5 x^{2}-7,\) find each function. $$ \text { If } P(x)=2 x-3, \text { find } P(a), P(-x), \text { and } P(x+h) $$

If \(f(x)=3 x+3, g(x)=4 x^{2}-6 x+3,\) and \(h(x)=5 x^{2}-7,\) find each function value. \(g(1)\)

In your own words, explain how to find the domain of a function given its graph.

Solve each equation for \(x\). $$ \frac{x-6}{4}=\frac{x-2}{5} $$
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