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Chapter 1: Problem 14

Find the slope and the \(y\) -intercept. $$ 2 x-y+3=0 $$

Short Answer

Expert verified
The slope is 2 and the y-intercept is 3.

Step by step solution

01

Isolate the y-variable

To find the slope and the y-intercept of the equation, rewrite it in the slope-intercept form, which is given by: \[y = mx + b\] where \(m\) is the slope and \(b\) is the y-intercept. Start by isolating the \(y\)-variable on one side. The given equation is: \[2x - y + 3 = 0\] Move all the terms involving \(x\) and constants to the other side of the equation: \[-y = -2x - 3\]
02

Solve for y

To get \(y\) on its own, multiply both sides by -1: \[y = 2x + 3\] Now the equation is in the slope-intercept form.
03

Identify the slope

In the slope-intercept form equation \(y = mx + b\), the coefficient of \(x\) represents the slope \(m\). Thus, for the equation \(y = 2x + 3\), the slope \(m = 2\).
04

Identify the y-intercept

In the slope-intercept form equation \(y = mx + b\), the constant term represents the y-intercept \(b\). Thus, for the equation \(y = 2x + 3\), the y-intercept \(b = 3\).

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Key Concepts

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

linear equations
Linear equations are fundamental in algebra and are used to describe a straight line on the coordinate plane. They are usually written in the form \(Ax + By = C\), but for our purposes, we'll focus on the slope-intercept form: \(y = mx + b\). Here, \(m\) is the slope and \(b\) is the y-intercept. This form makes it easy to identify these two important characteristics needed to draw the line.
Linear equations help describe relationships where variables change at a constant rate. In real-world contexts, they might model situations like calculating distances over time or predicting costs based on certain quantities.
finding slope
The slope of a line measures its steepness and direction. In the slope-intercept form equation \(y = mx + b\), the slope is represented by the coefficient \(m\). To find the slope, you need to rewrite an equation so that it's in this form.
In our exercise, the given equation is \(2x - y + 3 = 0\). By isolating \(y\) on one side, we rewrite it to: \(-y = -2x - 3\). Then, by multiplying both sides by -1, we get: \(y = 2x + 3\). Now, it's in slope-intercept form.
Here, you can see that the slope \(m = 2\). This means for every step you move right along the x-axis, you move up 2 units on the y-axis.
y-intercept
The y-intercept of a line is the point where the line crosses the y-axis. In the slope-intercept form equation \(y = mx + b\), the y-intercept is represented by the constant term \(b\). To find the y-intercept, make sure your equation is in slope-intercept form.
For example, in the equation \(y = 2x + 3\), the y-intercept \(b = 3\). This tells us that the line will cross the y-axis at the point (0, 3).
The y-intercept is useful for quickly graphing a line and understanding its position relative to the origin (0,0). It's also vital for interpreting real-world problems, like where a cost function meets initial setup charges without additional quantities.

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Most popular questions from this chapter

Sound Waves. The pitch of a sound wave is measured by its frequency. Humans can hear sounds in the range from 20 to \(20,000 \mathrm{~Hz}\), while dogs can hear sounds as high as \(40,000 \mathrm{~Hz}\). The loudness of the sound is determined by the amplitude. \({ }^{22}\). The note A above middle \(C\) on a piano generates a sound modeled by the function \(g(t)=4 \sin (880 \pi t)\), where \(t\) is in seconds. Find the frequency of \(\mathrm{A}\) above middle \(\mathrm{C}\).

Use a calculator to evaluate the following trigonometric functions. $$ \tan \left(-220^{\circ}\right) $$

For the following functions, find the amplitude, period, and mid-line. Also, find the maximum and minimum. $$ y=5 \cos (t / 2)+1 $$

While standing in the Mall in Washington, D.C., a tourist observes the angle of elevation to the top of the Washington Monument to be \(67^{\circ}\). After moving \(1012 \mathrm{ft}\) farther away from the Washington Monument, the angle of elevation changes to \(24^{\circ}\). a) Use the small triangle to find \(x\) in terms of \(h\). b) Use the large triangle to find the height of the Washington Monument.

Use a calculator to find the values of the following trigonometric functions. \(\sec (3 \pi / 8)\)
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