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Chapter 3: Problem 40

Write an equation of the line satisfying the given conditions. Give the final answer in slope-intercept form. (Hint: Recall the relationships among slopes of parallel and perpendicular lines.) Parallel to \(3 x+y=7\) \(y\) -intercept (0,4)

Short Answer

Expert verified
The equation of the line is \(y = -3x + 4\).

Step by step solution

01

Identify the slope of the given line

To find the slope of the line parallel to the given line, first write the equation of the line in slope-intercept form. The given line is \(3x + y = 7\). To convert this to slope-intercept form \(y = mx + b\), solve for \(y\).
02

Convert to slope-intercept form

Starting from \(3x + y = 7\), subtract \(3x\) from both sides to get \(y = -3x + 7\). The slope-intercept form (\(y = mx + b\)) reveals that the slope \(m\) of the given line is \(-3\).
03

Determine the slope for the line parallel

Since the lines are parallel, they have the same slope. Therefore, the slope \(m\) of the line we need to find is also \(-3\).
04

Use the y-intercept

We are given that the \(y\)-intercept of the line we need to find is \(4\). This means that the \(b\) in the slope-intercept form \(y = mx + b\) is \(4\).
05

Write the equation of the line

Now that we have the slope \(m = -3\) and the \(y\)-intercept \(b = 4\), we can write the equation of the line in slope-intercept form: \(y = -3x + 4\).

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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 crucial for understanding and graphing lines. It is written as \(y = mx + b\), where:
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Most popular questions from this chapter

Solve each problem. The weight \(y\) (in pounds) of a man taller than 60 in. can be approximated by the linear equation $$ y=5.5 x-220 $$ where \(x\) is the height of the man in inches. (a) Use the equation to approximate the weights of men whose heights are 62 in., 66 in., and 72 in. (b) Write the information from part (a) as three ordered pairs. (c) Graph the equation for \(x \geq 62\), using the data from part (b). (d) Use the graph to estimate the height of a man who weighs 155 lb. Then use the equation to find the height of this man to the nearest inch.

Find the \(x\) - and \(y\) -intercepts for the graph of each equation. $$ 2 x-3 y=0 $$

The height \(y\) (in centimeters) of a woman can be approximated by the linear equation $$ y=3.9 x+73.5 $$ where \(x\) is the length of her radius bone in centimeters. (a) Use the equation to approximate the heights of women with radius bones of lengths \(20 \mathrm{~cm}, 22 \mathrm{~cm},\) and \(26 \mathrm{~cm}\). (b) Write the information from part (a) as three ordered pairs. (c) Graph the equation for \(x \geq 20\), using the data from part (b). (d) Use the graph to estimate the length of the radius bone in a woman who is \(167 \mathrm{~cm}\) tall. Then use the equation to find the length of the radius bone to the nearest centimeter.

Write an equation of the line passing through the given pair of points. Give the final answer in (a) slope-intercept form and (b) standard form. (-1,-7) and (-8,-2)

Use the fact that \(m=-\frac{A}{B}\) to find the slope of the line with each equation. (a) \(2 x+3 y=18\) (b) \(4 x-2 y=-1\) (c) \(3 x-7 y=21\)
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