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Chapter 3: Problem 76

(a) find the \(y\)-intercept. (b) find the \(x\)-intercept. (c) use the slope formula to find the slope of the line. \(x-y=7\)

Short Answer

Expert verified
(0, -7); (7, 0); slope = 1

Step by step solution

01

Finding the y-intercept

To find the y-intercept, set the value of x to 0 in the equation and solve for y. Start with the equation: \(x - y = 7 \) Setting \(x = 0\), we get: \(0 - y = 7\) \(y = -7\) So, the y-intercept is \( (0, -7) \).
02

Finding the x-intercept

To find the x-intercept, set the value of y to 0 in the equation and solve for x. Starting with the equation: \(x - y = 7 \) Setting \(y = 0\), we get: \(x - 0 = 7\) \(x = 7\) So, the x-intercept is \( (7, 0) \).
03

Using the slope formula to find the slope

Rearrange the given equation into the slope-intercept form \(y = mx + c \), where \(m\) is the slope. Start with the original equation: \(x - y = 7 \) Rearrange it to solve for y: \(y = x - 7\) So, the slope \(m\) is 1.

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Key Concepts

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

Understanding the y-intercept
To find the y-intercept of a line, we look at where the line crosses the y-axis. This happens when the value of x is 0. In our equation, which is given as: \( x - y = 7 \) set \( x = 0 \) and solve for \( y \): \( 0 - y = 7 \) Simplify to get: \( y = -7 \) So, the y-intercept is the point \( (0, -7) \). The y-intercept is crucial because it shows where the line meets the y-axis. This is helpful for graphing the line or understanding its position relative to the y-axis.
Grasping the x-intercept
The x-intercept is the point where the line crosses the x-axis, which occurs when the value of y is 0. With our given equation: \( x - y = 7 \) we set \( y = 0 \) and solve for \( x \): \( x - 0 = 7 \) Simplify to find \( x \): \( x = 7 \) Hence, the x-intercept is \( (7, 0) \). Knowing the x-intercept is important as it tells us where the line meets the x-axis. This is particularly useful for understanding the geometry of the line.
Using the slope formula
The slope tells us how steep a line is. It shows the rate at which y changes with respect to x. We start with the general linear equation: \( x - y = 7 \) To find the slope, we rearrange this equation into the slope-intercept form, which is \( y = mx + c \), where \( m \) is the slope. First, solve for \( y \): \( x - y = 7 \) Rearrange to: \( y = x - 7 \) In this form, comparing it to \( y = mx + c \), we see \( m = 1 \). Therefore, the slope of our line is 1. Understanding the slope helps in determining the direction and steepness of the line.

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Most popular questions from this chapter

For exercises 1-8, (a) represent the information as two ordered pairs. (b) find the average rate of change, \(m\). The amount of certified organic cropland in Washington State planted in peas increased from 28 acres in 2007 to 252 acres in 2010. Round to the nearest whole number. (Source: www.tfrec.wsu.edu, March 2011)

For exercises 9鈥�20, (a) graph the given points, and draw a line through the points. (b) use the graph to find the slope of the line. (c) use the slope formula to find the slope of the line. \((1,4) ;(3,10)\)

Use the slope formula to find the slope of the line that passes through the points. \((-1,5) ;(-6,-13)\)

(a) find the \(y\)-intercept. (b) find the \(x\)-intercept. (c) use the slope formula to find the slope of the line. \(2 x+3 y=9\)

Solve. \(m=\frac{-10-4}{-5-(-3)}\)
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